ECSS-E-ST-60-20C Rev. 1

15 November 2008

Space engineering

Stars sensors terminology and

performance specification



ECSS Secretariat

ESA-ESTEC

Requirements & Standards Division

Noordwijk, The Netherlands

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Foreword

This Standard is one of the series of ECSS Standards intended to be applied together for the

management, engineering and product assurance in space projects and applications. ECSS is a

cooperative effort of the European Space Agency, national space agencies and European industry

associationsforthepurposeofdevelopingand

maintainingcommonstandards.Requirementsinthis

Standardaredefinedintermsofwhatshallbeaccomplished,ratherthanintermsofhowtoorganize

and perform the necessary work. This allows existing organizational structures and methods to be

appliedwherethey are effective, and for thestructuresand methods to evolve

 as necessarywithout

rewritingthestandards.

This Standard has been prepared by the ECSS‐E‐ST‐60‐20 Working Group, reviewed by the ECSS

ExecutiveSecretariatandapprovedbytheECSSTechnicalAuthority.

Disclaimer

ECSSdoesnotprovideanywarrantywhatsoever,whetherexpressed,implied,orstatutory,including,

butnotlimitedto,

anywarrantyofmerchantabilityorfitnessforaparticularpurposeoranywarranty

that the contents of the item are error‐free. In no respect shall ECSS incur any liability for any

damages,including,butnotlimitedto, direct, indirect,special,orconsequentialdamagesarisingout

of, resulting from, or

in any way connected to the use of this Standard, whether or not based upon

warranty, business agreement,tort, or otherwise; whether ornot injury wassustained bypersons or

propertyorotherwise;andwhetherornotlosswassustainedfrom,oraroseoutof,theresultsof,the

item,or

anyservicesthatmaybeprovidedbyECSS.

Publishedby: ESARequirementsandStandardsDivision

ESTEC, P.O. Box 299,

2200 AG Noordwijk

The Netherlands

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Change log

ECSS‐E‐ST‐60‐20A Neverissued

ECSS‐E‐ST‐60‐20B Neverissued

ECSS‐E‐ST‐60‐20C

31July2008

Firstissue

ECSS‐E‐ST‐60‐20CRev.1

15November2008

Firstissuerevision1.

ChangeswithrespecttoversionC(31July2008)areidentifiedwith

revisiontracking.

Mainchangesare:

Theterm“imaginaryensemble”hasbeenreplacedinthewholedocument

with“statisticalensemble”tobeconsistentwithECSS‐E

‐ST‐60‐10.

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Table of contents

1 Scope.......................................................................................................................9

2 Normative references...........................................................................................10

3 Terms, definitions and abbreviated terms..........................................................11

3.1 Terms from other standards .....................................................................................11

3.2 Terms specific to the present standard ....................................................................11

3.3 Abbreviated terms ....................................................................................................30

4 Functional requirements......................................................................................32

4.1 Star sensor capabilities ............................................................................................32

4.1.1 Overview.....................................................................................................32

4.1.2 Cartography................................................................................................33

4.1.3 Star tracking ...............................................................................................34

4.1.4 Autonomous star tracking...........................................................................34

4.1.5 Autonomous attitude determination............................................................35

4.1.6 Autonomous attitude tracking.....................................................................36

4.1.7 Angular rate measurement.........................................................................36

4.1.8 (Partial) image download............................................................................37

4.1.9 Sun survivability..........................................................................................37

4.2 Types of star sensors...............................................................................................38

4.2.1 Overview.....................................................................................................38

4.2.2 Star camera................................................................................................38

4.2.3 Star tracker.................................................................................................38

4.2.4 Autonomous star tracker ............................................................................38

4.3 Reference frames.....................................................................................................39

4.3.1 Overview.....................................................................................................39

4.3.2 Provisions...................................................................................................39

4.4 On-board star catalogue...........................................................................................39

5 Performance requirements..................................................................................41

5.1 Use of the statistical ensemble.................................................................................41

5.1.1 Overview.....................................................................................................41

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5.1.2 Provisions...................................................................................................41

5.2 Use of simulations in verification methods ...............................................................42

5.2.1 Overview.....................................................................................................42

5.2.2 Provisions for single star performances .....................................................42

5.2.3 Provisions for quaternion performances.....................................................42

5.3 Confidence level.......................................................................................................42

5.4 General performance conditions ..............................................................................43

5.5 General performance metrics...................................................................................44

5.5.1 Overview.....................................................................................................44

5.5.2 Bias.............................................................................................................44

5.5.3 Thermo elastic error ...................................................................................45

5.5.4 FOV spatial error ........................................................................................46

5.5.5 Pixel spatial error........................................................................................47

5.5.6 Temporal noise...........................................................................................47

5.5.7 Aberration of light .......................................................................................48

5.5.8 Measurement date error.............................................................................49

5.5.9 Measured output bandwidth .......................................................................49

5.6 Cartography..............................................................................................................49

5.7 Star tracking .............................................................................................................49

5.7.1 Additional performance conditions .............................................................49

5.7.2 Single star tracking maintenance probability..............................................50

5.8 Autonomous star tracking.........................................................................................50

5.8.1 Additional performance conditions .............................................................50

5.8.2 Multiple star tracking maintenance level.....................................................50

5.9 Autonomous attitude determination..........................................................................51

5.9.1 General.......................................................................................................51

5.9.2 Additional performance conditions .............................................................51

5.9.3 Verification methods...................................................................................52

5.9.4 Attitude determination probability ...............................................................52

5.10 Autonomous attitude tracking...................................................................................53

5.10.1 Additional performance conditions .............................................................53

5.10.2 Maintenance level of attitude tracking ........................................................54

5.10.3 Sensor settling time....................................................................................55

5.11 Angular rate measurement.......................................................................................55

5.11.1 Additional performance conditions .............................................................55

5.11.2 Verification methods...................................................................................55

5.12 Mathematical model .................................................................................................56

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Annex A (normative) Functional mathematical model (FMM) description -

DRD ......................................................................................................................57

Annex B (informative) Ancillary terms in Star Sensors .......................................59

Annex C (informative) Optional features of star sensors ....................................68

Annex D (informative) Performance metrics applied to star sensors.................72

Annex E (informative) Statistics.............................................................................74

Annex F (informative) Transformations between coordinate frames .................80

Annex G (informative) Contributing Error Sources..............................................82

Annex H (informative) Example of data sheet.......................................................84

Figures

Figure 3-1: Star sensor elements – schematic.......................................................................14

Figure 3-2: Example alignment reference frame....................................................................16

Figure 3-3: Boresight reference frame ...................................................................................17

Figure 3-4: Example of Inertial reference frame.....................................................................17

Figure 3-5: Mechanical reference frame ................................................................................18

Figure 3-6: Schematic illustration of reference frames...........................................................19

Figure 3-7: Stellar reference frame ........................................................................................19

Figure 3-8: Schematic timing diagram....................................................................................21

Figure 3-9: Field of View ........................................................................................................23

Figure 3-10: Aspect angle to planetary body or sun...............................................................24

Figure 4-1: Schematic generalized Star Sensor model..........................................................33

Figure B-1 : AME, MME schematic definition.........................................................................63

Figure B-2 : RME Schematic Definition..................................................................................64

Figure B-3 : MDE Schematic Definition..................................................................................65

Figure B-4 : Rotational and directional Error Geometry .........................................................66

Figure F-1 : Angle rotation sequence.....................................................................................81

Figure H-1 : Example of detailed data sheet..........................................................................85

Tables

Table C-1 : Minimum and optional capabilities for star sensors.............................................71

Table D-1 : Measurement error metrics .................................................................................73

Table D-2 : Star Position measurement error metrics............................................................73

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Table E-1 : Minimum number of simulations to verify a performance at performance

confidence level P

to an estimation confidence level of 95 %............................78

Table G-1 : Contributing error sources...................................................................................82

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Introduction

Inrecentyearstherehave been rapiddevelopmentsinstartrackertechnology,

in particular with a great increase in sensor autonomy and capabilities. This

Standard is intended to support the variety of star sensors either available or

underdevelopment.

This Standard defines the terminology and specification definitions for the

performanceofstartrackers(inparticular,autonomousstartrackers).Itfocuses



on the specific issues involved in the specification of performances of star

trackersandisintendedtobeusedasastructuredsetofsystematicprovisions.

This Standard is not intended to replace textbook material on star tracker

technology,andsuchmaterialisintentionallyavoided.Thereadersandusersof

this Standard are assumed to possess general knowledge of star tracker

technologyanditsapplicationtospacemissions.

This document defines and normalizes terms used in star sensor performance

specifications,aswellassomeperformanceassessmentconditions:

• sensorcomponents

• sensorcapabilities

• sensortypes

• sensorreferenceframes

•

sensormetrics



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Scope

ThisStandardspecifiesstartrackerperformancesaspartofaspaceproject.The

Standard covers all aspects of performances, including nomenclature,

definitions, and performance metrics for the performance specification of star

sensors.

The Standard focuses on performancespecifications.Other specificationtypes,

for example mass and power, housekeeping data, TM/TC interface and

 data

structures,areoutsidethescopeofthisStandard.

When viewed from the perspective of a specific project context, the

requirementsdefinedinthisStandardshouldbetailoredto match thegenuine

requirementsofaparticularprofileandcircumstancesofaproject.

Thisstandardmaybetailoredforthespecific

characteristicsandconstraintsofa

spaceprojectinconformancewithECSS‐S‐ST‐00.



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Normative references

The following normative documents contain provisions which, through

reference in this text, constitute provisions of this ECSS Standard. For dated

references,subsequentamendmentsto,orrevisionofanyofthesepublications,

donotapply.However,partiestoagreementsbasedonthisECSSStandardare

encouragedtoinvestigatethepossibilityofapplying

themorerecenteditionsof

the normative documents indicated below. For undated references, the latest

editionofthepublicationreferredtoapplies.



ECSS‐S‐ST‐00‐01 ECSSsystem–Glossaryofterms



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Terms, definitions and abbreviated terms

3.1 Terms from other standards

ForthepurposeofthisStandard,thetermsanddefinitionsfromECSS‐S‐ST‐00‐01

apply.Additionaldefinitionsareincludedin

AnnexB.

3.2 Terms specific to the present standard

3.2.1 Capabilities

3.2.1.1 aided tracking

capability to input information to the star sensor internal processing from an

externalsource

NOTE1 This capability applies to star tracking,

autonomous star tracking and autonomous

attitudetracking.

NOTE2 E.g.AOCS.

3.2.1.2 angular rate measurement

capability to determine, the instantaneous sensor reference frame inertial

angularrotationalrates

NOTE Angularratecanbecomputedfromsuccessivestar

positions obtained from the detector or successive

absolute attitude (derivation of successive

attitude).

3.2.1.3 autonomous attitude determination

capability to determine the absolute orientation of a defined sensor reference

frame with respect to a defined inertial reference frame and to do so without

the use of any a priori or externally supplied attitude, angularrate or angular

accelerationinformation

3.2.1.4 autonomous attitude tracking

capabilitytorepeatedlyre‐assessandupdatetheorientationofasensor‐defined

reference frame with respect to an inertially defined reference frame for an

extended period of time, using autonomously selected star images in the field

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of view, following the changing orientation of the sensor reference frame as it

movesinspace

NOTE1 TheAutonomousAttitudeTrackingmakesuseofa

supplied a priori Attitude Quaternion, either

provided by an external source (e.g. AOCS) or as

the output of an Autonomous Attitude

Determination(‘Lost‐in‐Space’

solution).

NOTE2 The autonomous attitude tracking functionality

can also be achieved by the repeated use of the

AutonomousAttitudeDeterminationcapability.

NOTE3 The Autonomous Attitude Tracking capability

does not imply the solution of the ‘lost in space’

problem.

3.2.1.5 autonomous star tracking

capabilitytodetect,locate,selectandsubsequentlytrackstarimageswithinthe

sensorfieldofview foranextended period oftimewith no assistanceexternal

tothesensor

NOTE1 Furthermore, the autonomous star tracking

capability is taken to include the ability to

determinewhenatrackedimageleavesthe

sensor

fieldofviewand select a replacementimage to be

trackedwithoutanyuserintervention.

NOTE2 Seealso

3.2.1.9(startracking).

3.2.1.6 cartography

capability to scan the entire sensor field of view and to locate and output the

positionofeachstarimagewithinthatfieldofview

3.2.1.7 image download

capabilitytocapturethesignalsfromthedetectorovertheentiredetectorField

of view, at one instant (i.e. within a single integration), and output all of that

informationtotheuser

NOTE Seealso

3.2.1.8(partialimagedownload).

3.2.1.8 partial image download

capabilitytocapturethesignalsfromthedetectorovertheentiredetectorField

ofview,atone instant(i.e.withinasingleintegration),andoutput part ofthat

informationtotheuser

NOTE1 Partial image download is an image downloads

(see

3.2.1.7)whereonlyapartofthedetectorfield

of view can be output for any given specific

‘instant’.

NOTE2 Partial readout of the detector array (windowing)

andoutput of thecorrespondingpixelsignalsalso

fulfilthefunctionality.

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3.2.1.9 star tracking

capability to measure the location of selected star images on a detector, to

output the co‐ordinates of those star images with respect to a sensor defined

reference frame and to repeatedly re‐assess and update those co‐ordinates for

an extended period of time, following the motion of each image

 across the

detector

3.2.1.10 sun survivability

capability to withstand direct sun illumination along the boresight axis for a

certain period of time without permanent damage or subsequentperformance

degradation

NOTE This capability could be extended to flare

capability considering the potential effect of the

earthorthemoonintheFOV.

3.2.2 Star sensor components

3.2.2.1 Overview

Figure3‐1showsaschemeoftheinterfaceamongthegeneralized components

specifiedinthisStandard.

NOTE Usedasacamerathesensoroutputcanbelocated

directlyafterthepre‐processingblock.

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BAFFLE

OPTICAL

HEAD

OPTICAL SYSTEM

PROCESSOR

PROCESS OUPUT

DETECTOR

MEMORY

CAMERA

OUTPUT

PRE-PROCESSING

Figure3‐1:Starsensorelements–schematic

3.2.2.2 baffle

passive structure used to prevent or reduce the entry into the sensor lens or

aperture of any signals originating from outside of the

field of view of the

sensor

NOTE Baffle design is usually mission specific and

usually determines the effective exclusion angles

forthelimboftheEarth,MoonandSun.TheBaffle

canbemounted directly onthesensoror can be a

totally separate element. In the latter case, a

positioningspecificationwithrespecttothesensor

isused.

3.2.2.3 detector

element of the star sensor that converts the incoming signal (photons) into an

electricalsignal

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NOTE Usual technologies in use are CCD (charge

coupled device) and APS (active pixel sensor)

arrays though photomultipliers and various other

technologiescanalsobeused.

3.2.2.4 electronic processing unit

setoffunctionsofthesensornotcontainedwithintheopticalhead

NOTE Specifically,thesensorelectronicscontains:

• sensorprocessor;

• powerconditioning;

• softwarealgorithms;

• onboardstarcatalogue(ifpresent).

3.2.2.5 optical head

partofthesensorresponsibleforthecaptureandmeasurementoftheincoming

signal

NOTE Assuchitconsistsof

• theopticalsystem;

• thedetector(includinganycoolingequipment);

• the proximity electronics (usually detector

control, readout and interface, and optionally

pixelpre‐processing);

• themechanicalstructure

tosupporttheabove.

3.2.2.6 optical system

systemthat comprises the componentparts tocapture andfocusthe incoming

photons

NOTE Usually this consists of a number of lenses, or

mirrorsandfilters,andthesupportingmechanical

structure,stops,pinholesandslitsifused.

3.2.3 Reference frames

3.2.3.1 alignment reference frame (ARF)

referenceframefixedwithrespecttothesensorexternalopticalcubewherethe

origin of the ARF is defined unambiguously with reference to the sensor

externalopticalcube

NOTE1 The X‐, Y‐ and Z‐axes of the ARF are a right‐

handed orthogonal set of axes which are defined



unambiguously with respect to the normal of the

faces of the external optical cube.

Figure 3‐2

schematicallyillustratesthedefinitionoftheARF.

NOTE2 The ARF is the frame used to align the sensor

duringintegration.

NOTE3 This definition does not attempt to prescribe a

definitionoftheARF,otherthanitisaframefixed

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relative to the physical geometry of the sensor

opticalcube.

NOTE4 If the optical cube’s faces are not perfectly

orthogonal, the X‐axis can be defined as the

projection of the normal of the X‐facein the plane

orthogonaltotheZ‐axis,and the Y‐axis completes

the

RHS.

Optical

Cube

ARF

Sensor

Figure3‐2:Examplealignmentreferenceframe

3.2.3.2 boresight reference frame (BRF)

referenceframewhere:

• the origin of the Boresight Reference Frame (BRF)is defined

unambiguously with reference to the mounting interface plane of the

sensorOpticalHead;

NOTE In an ideally aligned opto‐electrical system this

results in a measured position at the centre of the

detector.

• the Z‐

axis of the BRF is defined to beanti‐parallel to the direction of an

incomingcollimatedlightraywhichisparalleltotheopticalaxis;

• X‐BRF‐axis isin the plane spanned by Z‐BRF‐axis and the vector from

the detector centre pointing along the positively counted

 detector rows,

as the axis perpendicular to Z‐BRF‐axis. The Y‐BRF‐axis completes the

righthandedorthogonalsystem.

NOTE1 TheX‐axesandY‐axesoftheBRFaredefinedtolie

(nominally) in the plane of the detector

perpendicular to the Z‐axis, so as to

form a right

handed set with one axis nominally along the

detector array row and the other nominally along

thedetectorarraycolumn.

Figure3‐3schematically

illustratesthedefinitionoftheBRF.

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NOTE2 The definition of the Boresight Reference Frame

does not imply that it is fixed with respect to the

Detector, but that it is fixed with respect to the

combineddetectorandopticalsystem.

Optics

Detector

BRF

Incoming light ray that

will give a measured

position at the centre of

the Detector.

Figure3‐3:Boresightreferenceframe

3.2.3.3 inertial reference frame (IRF)

referenceframedeterminedtoprovideaninertialreference

NOTE1 E.g.usetheJ2000referenceframeasIRFasshown

in

Figure3‐4.

NOTE2 TheJ2000referenceframe

(inshortforICRF–Inertial

Celestial Reference Frame at J2000 Julian date)

is

usually defined as Z IRF = earth axis of rotation

(direction of north) at J2000 (01/01/2000 at noon

GMT),XIRF=directionofvernalequinoxatJ2000,

Y IRF completes the right‐handed orthonormal

referenceframe.

Ecliptic Plane

Equatorial Plane

IRF

at J2000 Julian date

X-axis in direction of

vernal equinox

Earth

Figure3‐4:ExampleofInertialreferenceframe

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3.2.3.4 mechanical reference frame (MRF)

reference frame where the origin of the MRF is defined unambiguously with

referencetothemountinginterfaceplaneofthesensorOpticalHead

NOTE1 For Fused  Multiple Optical Head configurations,

the interface plane of one of the Optical Heads

may be nominated to define the MRF. The

orientationisto

bedefined.

NOTE2 E.g. the Z‐axis of the MRF is defined to be

perpendiculartothemountinginterfaceplane.The

X‐andY‐axesof the MRF are defined tolieinthe

mounting plane such as to form an orthogonal

RHSwiththeMRFZ‐axis.

NOTE

3 Figure 3‐5 schematically illustrates the definition

oftheMRF.

MRF

Spacecraft Body

Mounting Interface

MRF

Figure3‐5:Mechanicalreferenceframe

3.2.3.5 stellar reference frame (SRF)

reference frame for each star where the origin of any SRF is defined to be

coincidentwiththeBoresightReferenceFrame(BRF)origin

NOTE1 TheZ‐axisofanySRFisdefinedtobethedirection

from the SRF origin to the true position of the

selected star

Figure 3‐6 gives a schematic

representation of the reference frames.

Figure 3‐7

schematicallyillustratesthedefinitionoftheSRF.

NOTE2 The X‐ and Y‐ axes ofthe SRFare obtained under

the assumption that the BRF can be brought into

coincidencewiththeSRFbytworotations,thefirst

aroundthe BRF X‐axisandthesecond around the



newBRFY‐axis (which iscoincidentwith the SRF

Y‐axis).

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MRF

Optical

Cube

Spacecraft Body

BRF

Sensor

BRF

ARF

SRF

Mounting Plate

IRF Axes

Figure3‐6:Schematicillustrationofreferenceframes



SRF

BRF

Detector

Selected star

SRF

BRF

rotation

Figure3‐7:Stellarreferenceframe

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3.2.4 Definitions related to time and frequency

3.2.4.1 integration time

exposuretimeoverwhichphotonswerecollectedinthedetectorarraypriorto

readoutandprocessingtogeneratetheoutput(starpositionsorattitude)

NOTE1 Integrationtime can befixed, manually adjustable

orautonomouslyset.

NOTE2

Figure 3‐8 illustrates schematically the various

times defined together with their inter‐

relationship.Thefigureincludesdatabeingoutput

from two Optical Heads, each of which is

separately processed prior to generation of the

sensor output. Note that for a Fused Multiple

Optical Head sensor; conceptually it is assumed

that

 the filtered output is achieved via sequential

processingof datafrom a single head at a time as

the data is received.Hence, with this

understanding, the figure and the associated time

definitionsalsoapplytothissensorconfiguration.

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Integration

Processing

Output

Integration time

Optical

Head 1

Sample Time

Latency

Time data is

first available

PROCESSING

Optical Head

PROCESSING

OUT

Data is

accessed

OUT

Data

Flow

Time

Figure3‐8:Schematictimingdiagram

3.2.4.2 measurement date

dateoftheprovidedmeasurement

NOTE1 Incaseofonboardfilteringthemeasurementdate

candeviatefromindividualmeasurementdates.

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NOTE2 Usually the mid‐point of the integration time is

considered as measurement date for CCD

technology.

3.2.4.3 output bandwidth

maximumfrequencycontainedwithinthesensoroutputs

NOTE1 The bandwidth of the sensor is limited in general

byseveralfactors,including:

• integrationtime;

• samplingfrequency;

• attitudeprocessingrate;

• onboard filtering of data (in particular for

multipleheadunits).

NOTE2 The output bandwidth corresponds to the



bandwidthofthesensorseenasalow‐passfilter.

3.2.5 Field of view

3.2.5.1 half-rectangular field of view

angular region around the Boresight Reference Frame (BRF) frame Z‐axis,

specifiedbytheangularexcursionsaroundtheBRFX‐andY‐axesbetweenthe

BRFZ‐axisandtheappropriaterectangleedge,withinwhichastarproducesan

imageontheDetectorarraythatisthenusedbythestar

sensor

NOTE1 ThisFieldofViewisdeterminedbytheopticsand

Detectordesign.Thisisschematicallyillustratedin

Figure3‐9.

NOTE2 In the corners, the extent of the FOV for this

definition exceeds the quoted value (see

Figure

3‐9).

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Detector

BRF Z axis

Light cone for Full

Cone Field of View

Light cone for

Half-Rectangular

Field of View

Full Cone Field

of View

Half Rectangular

Field of View

Figure3‐9:FieldofView

3.2.5.2 full cone field of view

angular region around the Boresight Reference Frame (BRF) frame Z‐axis,

specifiedasafullconeangle,withinwhichastarwillproduceanimageonthe

Detectorarraythatisthenusedbythestarsensor

NOTE ThisFieldofViewisdeterminedbytheopticsand

Detectordesign.

Thisisschematicallyillustratedin

Figure3‐9.

3.2.5.3 pixel field of view

anglesubtendedbyasingleDetectorelement

NOTE PixelFieldofViewreplaces(andisidenticalto)the

commonlyusedtermInstantaneousFieldofView.

3.2.6 Angles of celestial bodies

3.2.6.1 aspect angle

half‐cone angle between the Boresight Reference Frame (BRF) Z‐axis and the

nearestlimbofacelestialbody

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Solar System

Body

Detector

BRF

SPECT ANGL

(in plane of diagram)

MEA

SEA

EEA

Figure3‐10:Aspectangletoplanetarybodyorsun

3.2.6.2 exclusion angle (EA)

lowestaspectangleofabodyatwhichquotedfullperformanceisachieved

NOTE1 The following particular exclusion angles can be

considered:

• TheEarthexclusionangle(EEA),definedasthe

lowest aspect angle of fully illuminated Earth

(including the Earth atmosphere) at which

quoted full performance is achieved, as

 shown

schematicallyin

Figure3‐10.

• The SunExclusionAngle (SEA),definedas the

lowest Aspect Angle of the Sun at which

quoted full performance is achieved, as shown

schematicallyin

Figure3‐10.

• TheMoonExclusionAngle(MEA)isdefinedas

the lowest Aspect Angle of the Full Moon at

which quoted full performance is achieved, as

shownschematicallyin

Figure3‐10.

NOTE2 The value of any EA depends on the distance to

the object. In general, the bandwidth is the lowest

of the cut‐off frequencies implied by the above

factors.

3.2.7 Most common terms

3.2.7.1 correct attitude

attitude for which the quaternion absolute measurementerror (AMEq defined

in

D.2.2)islowerthanagiventhreshold

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3.2.7.2 correct attitude threshold

maximum quaternion absolute measurement error (AMEq) for which an

attitudeisacorrectattitude

3.2.7.3 false attitude

attitudewhichisanoncorrectattitude

3.2.7.4 false star

signal on the detector not arising from a stellar source but otherwise

indistinguishablefromastarimage

NOTE This definition explicitly excludes effects from the

Moon, low incidence angle proton effects etc.,

whichcangenerallybedistinguishedasnon‐stellar

inoriginbygeometry.

3.2.7.5 image output time

timerequiredtooutputthedetectorimage

3.2.7.6 statistical ensemble

setofsensors(notallactuallybuilt)onwhichtheperformancesareassessedby

useofstatisticaltoolsonasetofobservationsandobservationconditions

NOTE1 The statistical

 ensemble is defined on a case‐by‐

case basis, depending on the performances to be

assessed.

NOTE2 See

5.1andAnnexEforfurtherdetails.

3.2.7.7 maintenance level of attitude tracking

total time within a longerdefinedintervalthat attitude tracking is maintained

(i.e. without any attitude acquisition being performed) with a probability of

100%foranyinitialpointingwithinthecelestialsphere

NOTE ThisparametercanalsobespecifiedasMeanTime

between loss of tracking or probability to

loose

trackingpertimeunit.

3.2.7.8 multiple star tracking maintenance level

total time within a longer defined interval that at least ‘n’ star tracks are

maintainedwithaprobabilityof100%

NOTE ThiscoversthecasewherethestarsintheFOVare

changing, such that the star tracks maintained

evolvewithtime.

3.2.7.9 night sky test

testperformedduringnighttimeusingtheskyasphysicalstimulusforthestar

sensor. The effect of atmospheric extinction should be taken into account and

reducedbyappropriatechoiceofthelocationfortest

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3.2.7.10 probability of correct attitude determination

probability that a correct attitude solution is obtained and is flagged as valid,

within a defined time from the start of attitude determination with the sensor

switchedonandattheoperatingtemperature

NOTE1 Time periods for other conditions, like recovery

aftertheSunenteringthe FOV or acold

start, can

be defined as the time needed to reach the start

time of the attitude determination. The total time

neededwouldthenbethesumofthetimeneeded

toreachthestarttimeoftheattitudedetermination

andthetimeperiodrelatedtothismetric.

NOTE2 Attitude

solution flagged as valid means that the

obtained attitude is considered by star sensor

suitable for use by the AOCS. The validity is

independentofaccuracy.

NOTE3 Correct attitude solution means that stars used to

derive the quaternion have been correctly

identified, i.e. error on delivered measurement is

belowa

definedthreshold.

3.2.7.11 probability of false attitude determination

probability that not correct attitude solution is obtained, which is flagged as

valid, within a defined time from the start of attitude determination with the

sensorswitchedonandattheoperatingtemperature

3.2.7.12 probability of invalid attitude solution

probabilitythatanattitudesolution(correctornotcorrect)isobtainedanditis

flagged as not valid, within a defined time from the start of attitude

determinationwiththesensorswitchedonandattheoperatingtemperature

NOTE1 The value of the Probability of Invalid Attitude

Solution is 1

‐(Probability of Correct Attitude

Determination + Probability of False Attitude

Determination).

NOTE2 Invalid attitude solutions include cases of silence

(i.e.noattitudeisavailablefromstarsensor).

3.2.7.13 sensor settling time

time period from the first quaternion output to the first quaternion at full

attitude accuracy, for random initial pointing within a defined region of the

celestialsphere

NOTE The time period is specified with a probability of

n%‐ifnotquoted,avalueof99%isassumed.

3.2.7.14 single star tracking maintenance probability

probabilitytobemaintainedbyanexistingstartrackoveradefinedtimeperiod

whilethetrackedstarisintheFOV

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3.2.7.15 star image

patternoflightfallingonthedetectorfromastellarsource

3.2.7.16 star magnitude

magnitudeofthestellarimageasseenbythesensor

NOTE Star magnitude takes into account spectral

considerations. This is also referred to as

instrumentalmagnitude.

3.2.7.17 validity

characteristics of an output of the star sensor being accurate enough for the

purposeitisintendedfor

NOTE E.g.usebytheAOCS.

3.2.8 Errors

3.2.8.1 aberration of light

Erroronthepositionofameasuredstarduetothetimeofpropagationoflight,

andthelinearmotionoftheSTRinaninertialcoordinatesystem

NOTE1 The Newtonian first order expression of the

rotationerrorforonestardirectionis:

()

usin

θ=ε



where:

V isthemagnitudeoftheabsolutelinearvelocity

ofthespacecraftw.r.t.toaninertialframe

c isthelightvelocity(299792458m/s)

 istheanglebetweenthe V

vectorandthestar

direction



∧



NOTE2 For a satellite on an orbit around the Earth, the

absolute velocity is the vector sum of the relative

velocityofthespacecraftw.r.ttheEarthandofthe

velocityoftheEarthw.r.ttheSun.

NOTE3 For an Earth orbit, the magnitude of this effect is

around 25 arcsec (max). For an interplanetary

spacecraft the absolute velocity is simply the

absolutevelocityw.r.t.thesun.

NOTE4 The associated metrics is the MDE (see Annex

B.5.11 for the mathematical definition). The

detailed contributors to the relativistic error are

givenin

AnnexG.

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3.2.8.2 bias

errorontheknowledgeoftheorientationoftheBRFincluding:

• the initial alignment measurement error between the Alignment

ReferenceFrame(ARF)andthe sensorBoresightReferenceFrame(BRF)

(ongroundcalibration)

• theAlignmentStability Error(CalibrationtoFlight)witchisthechange

in the transformation between the sensor

 Mechanical Reference Frame

(MRF)andthesensorBoresightReferenceFrame(BRF)betweenthetime

ofcalibrationandthestartofthein‐flightmission

NOTE1 The bias can be for the BRF Z‐axis directional or

therotationalerrorsaroundtheBRFX,Y‐axes.

NOTE2 For definition

of directional and rotational errors

see

B.5.14andB.5.17.

NOTE3 Duetoitsnature,thebiasmetricvalueisthesame

whatevertheobservationareais.

NOTE4 The associated metrics is the MME (see Annex

B.5.7forthemathematicaldefinition).Thedetailed

contributorstothebiasaregivenin

AnnexG.

3.2.8.3 FOV spatial error

error on the measured attitude quaternion due to theindividual spatial errors

onthestars

NOTE1 This error has a spatial periodicity, whose

amplitude is defined by the supplier. It ranges

fromafewpixelsuptothefullcameraFOV.

NOTE2 FOV spatial errors are mainly due to

 optical

distortion. These errors can be converted to time

domain using sensor angular rate. Then, from

temporalfrequencypointofview,theyrangefrom 

bias to high frequency errors depending on the

motion of stars on the detector. They lead to bias

error in the case of inertial pointing, while

they

contribute to random noise for high angular rate

missions.

NOTE3 The associated metrics is the MDE (see Annex

B.5.11 for the mathematical definition). The

detailed contributors to the FOV spatial error are

givenin

AnnexG.

3.2.8.4 pixel spatial error

Measurement errors of star positions due to detector spatial non uniformities

(including PRNU, DSNU, dark current spikes, FPN) and star centroid

computation(alsocalledinterpolationerror)

NOTE1 Becauseoftheir‘spatial’nature–theseerrorsvary

withthepositionofstarsonthedetector–theyare

well captured by

 metrics working in the angular

domain. The pixel spatial errors are then well

defined as the errors on the measured attitude

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(respectively the measured star positions) due to

star measurement errors with spatial period of

TBD angular value. Several classes of spatial

periodscanbeconsidered.

NOTE2 These errors can be converted to time domain

using sensor angular rate. Then, from temporal

frequency point of view, they range from bias to



highfrequencyerrors dependingonthemotionof

starsonthedetector.Theyleadtobiaserrorinthe

case of inertial pointing, while they contribute to

randomnoiseforhighangularratemissions.

NOTE3 The associated metrics is the MDE (see Annex

B.5.11 for the mathematical definition). The

detailed contributors to the pixel spatial error are

givenin

AnnexG.

3.2.8.5 temporal noise

Temporal fluctuation on the measured quaternion (star positions) due to time

variationerrorsources

NOTE1 Temporalnoiseisawhitenoise.

NOTE2 TheassociatedmetricsistheRME(seeAnnex

B.5.8

for the mathematical definition). The detailed

contributors to the temporal noise error are given

in

AnnexG.

3.2.8.6 thermo elastic error

deviation of BRFversus MRF for a given temperature variation ofthe

mechanical interface of the optical head of the sensor and thermal power

exchangewithspace

NOTE1 The detailed contributors to the thermo elastic

erroraregivenin

AnnexG.

NOTE2 The associated metrics is the MDE (see Annex

B.5.11 for the mathematical definition). FOV

spatialerror.

3.2.9 Star sensor configurations

3.2.9.1 fused multiple optical head configuration

more than one Optical Head, each with a Baffle, and a single Electronic

ProcessingUnitproducingasinglesetofoutputsthatusesdatafromallOptical

Heads

3.2.9.2 independent multiple optical head configuration

more than one optical head, each with a  baffle, and a single electronic

processingunitproducingindependentoutputsforeachopticalhead

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3.2.9.3 integrated single optical head configuration

singleopticalheadplusbaffleandasingleelectronicprocessingunitcontained

withinthesamemechanicalstructure

3.2.9.4 separated single optical head configuration

singleopticalheadplusbaffleandasingleelectronicprocessingunitwhichare

notcollocatedwithinthesamemechanicalstructure

3.3 Abbreviated terms

For the purpose of this Standard, theabbreviated terms from ECSS‐S‐ST‐00‐01

andthefollowingapply:

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Abbreviation Meaning

AME

absolutemeasurementerror

APS

activepixelsensor

ARF

alignmentreferenceframe

ARME

absoluteratemeasurementerror

AST

autonomousstartracker

BRF

boresightreferenceframe

BOL

beginning‐of‐life

CCD

chargecoupleddevice

CTE

chargetransferefficiency

DSNU

darksignalnon‐uniformity

EEA

Earthexclusionangle

EOL

end‐of‐life

FMM



functionalmathematicalmodel

FOV

fieldofview

FPN

fixpatternnoise

GRME

generalizedrelativemeasurement

error

IRF

inertialreferenceframe

LOS

lineofsight

MDE

measurementdrifterror

MEA

Moonexclusionangle

MME

meanmeasurementerror

MRE

measurementreproducibilityerror

MRF

mechanicalreferenceframe

PRNU

photoresponsenon‐uniformity

RME

relativemeasurementerror

RHS

righthandedsystem

SEA

Sunexclusionangle

SEU

singeeventupset

SET

singleeventtransient

SRF

stellarreferenceframe

STC

starcamera

STM

starmapper

STR

startracker

STS

starscanner

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Functional requirements

4.1 Star sensor capabilities

4.1.1 Overview

This subclause describes the different main capabilities of star sensors. These

capabilities are defined with respect to a generalized description of the

reference frames (either sensor‐referenced or inertially referenced in clause

3).

This set of capabilities is then later used to describe the specific types of star

sensorandtheirperformances.

In order to describe the star sensor capabilities, the following generalized

sensormodelisused:

A star sensor comprises an imaging function, a detecting function and a data

processingfunction. The

 imaging function collects photons from objects in the

field of view of the sensor and focuses them on a detecting element. This

element converts the photons into an electrical signal that is then subject to

someprocessingtoproducethesensoroutput.

Aschematicofthissensormodelispresented

inFigure4‐1.

For each capability the nominal outputs and additional outputs are defined.

Thesefunctionaldatashouldbeidentifiedinthetelemetrylistcomingfromthe

starsensor.

Theoutputsasdefinedinthisdocumentarepurelyrelatedtotheperformance

of the sensor, and represent the minimum information to

 be provided by the

sensor to possess the capability. Other aspects, such as sensor housekeeping

data, data structures and the TM/TC interface, are outside the scope of this

Standard.

NOTE1 The same capabilities can be defined for Star

Sensors employed on spinning spacecraft (Star

Scanner)wherestarimagesareacquiredatangular

rateuptotensofdeg/sdrivingthedetectorwitha

dedicated technique. For Star Sensor based on

CCD detector, an example of this technique could

be the Time Delay Integration (TDI). It is outside

the scope of this specification to give detailed

capabilitydefinitionsforthiskindofsensor.

NOTE2 OptionalfeaturesareincludedinAnnex

B.6.

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Star

Light from star

Optics

Detector

Processor

Processed

output

Figure4‐1:SchematicgeneralizedStarSensormodel

4.1.2 Cartography

4.1.2.1 Inputs

a. Theacquisitioncommandshallbesuppliedasminimumsetofinputs.

4.1.2.2 Outputs

a. Asensorwithcartographycapabilityshallhavethefollowingminimum

outputs:

1. starposition,

2. measurementdate.

b. WhentheStarImageismeasuredinaDetector‐fixedframewhichisnot

the same as the Boresight Reference Frame (BRF), the output shall be

convertedintotheBoresightReferenceFrame

(BRF).

NOTE The output parameterization is the Star Image

position in the Boresight Reference Frame (BRF),

givenbythetwomeasuresoftheangularrotations

which define the transformation from the BRF to

thestarStellarReferenceFrame(SRF).

c. The date of measurement shall be expressed as a (scalar)

 number

indicating the delay relative to a known external time reference agreed

withthecustomer.

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4.1.3 Star tracking

4.1.3.1 Inputs

a. The minimum set of inputs to be supplied in order to initializethe Star

Trackingshallbe:

1. theinitialstarposition;

2. theangularrate;

3. validitydate.

b. For aided tracking, data specified in

4.1.3.1ashall be supplied regularly

bythespacecraft,atanupdaterateandaccuracyagreedbythecustomer.

c. Theunitofallinputsshallbeindicated.

4.1.3.2 Outputs

a. A sensor with the star tracking capability shall have the following

minimumoutputs:

1. the position of each Star Image with respect to a sensor‐defined

referenceframe;

2. focallengthifstarpositiononthedetectorchipisoutputinunits

oflength;

3. themeasurementdate.

NOTE1

 Theinitialselectionofthestarimagestobetracked

bythesensorisnotincludedwithinthiscapability

and sometimes cannot be done without assistance

externaltothesensor.

NOTE2 The output parameterization is the Star Image

position in the Boresight Reference Frame (BRF),

givenbythetwo

measuresoftheangularrotations

[

]

MEASYMEASXMEAS

sss

 which define the

transformation from the BRF to the star Stellar

ReferenceFrame(SRF).

NOTE3 This capability does not imply to autonomously

identifythestarimagesasimagestobetracked  or

explicitly identified by the unit. However, it does

includetheabilitytomaintaintheidentificationof

each

star image and to correctly update the co‐

ordinates of each image as it moves across the

detectorduetotheangularrateofthesensor.

4.1.4 Autonomous star tracking

4.1.4.1 Inputs

a. The minimum set of inputs to be supplied in order to initialize the

AutonomousStarTrackingshallbe:

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1. theangularrate;

2. thevaliditydate.

b. For aided tracking, data specified in

4.1.4.1ashall be supplied regularly

bythespacecraft,atanupdaterateandaccuracyagreedbythecustomer.

c. Theunitofallinputsshallbeindicated.

4.1.4.2 Outputs

a. A sensor with the autonomous star tracking capability shall have the

minimumoutputs:

1. the position of each star image with respect to a sensor‐defined

referenceframe;

2. theMeasurementdate.

NOTE This capability does not imply the stars to be

explicitly identified by the unit. However, it

does

includetheabilitytomaintaintheidentificationof

each star image once selected, to correctly update

the co‐ordinates of each image as it moves across

the detector, and autonomouslymanagethe set of

starimagesbeingtracked.

4.1.5 Autonomous attitude determination

4.1.5.1 Inputs

a. Theacquisitioncommandshallbesuppliedasaminimumsetofinputs.

NOTE When a priori initial attitude information for

example an initial quaternion or a restriction

within the celestial sphere, is supplied by the

ground the capability is referred as Assisted

Attitudedetermination

4.1.5.2 Outputs

a. A sensor with autonomous attitude determination shall have the

minimumoutputs:

1. therelativeorientationofthedefinedsensor referenceframewith

respecttothedefinedinertialreferenceframe;

NOTE Therelativeorientationisusuallyexpressedinthe

formofanormalizedattitudequaternion

2. theMeasurementdate;

a validity index or flag estimating the validity of the determined

attitude.

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4.1.6 Autonomous attitude tracking

4.1.6.1 Inputs

a. The minimum set of inputs to be supplied in order to initialize the

AutonomousAttitudeTrackingshallbe:

1. theattitudequaternion;

2. the3‐dimensionangularrate vector givingthe angularrate of the

sensorBRFwithrespecttotheIRF;

NOTE Thisvectorisexpressedinthe

sensorBRF.

3. thevaliditydateforbothsuppliedattitudeandangularrate.

b. For aided tracking, data specified in

4.1.6.1ashall be supplied regularly

bythespacecraft,atanupdaterateandaccuracyagreedbythecustomer.

c. Exceptforattitudequaternion,theunitofallinputsshallbeindicated.

d. The supplier shall document whether the star sensor initialization uses

either:

⎯ Internalinitialization,or

NOTE Theinformationtoinitializethesensorisprovided

by the attitude determination function of the star

sensor.

⎯ Directinitialization.

NOTE Theinformationtoinitializethesensorissupplied

byanexternalsourcee.g.AOCS.

4.1.6.2 Outputs

a. A sensor with autonomous attitude tracking capability shall have the

followingminimumoutputs:

1. the orientation of the sensor defined reference frame with respect

tothe inertiallydefinedreferenceframe(nominallyin the form of

anattitudequaternion);

2. theMeasurementdate;

3. a validity index or flag, estimating the

 validity of the determined

attitude;

4. measurementofStarMagnitudeforeachtrackedStarImage.

4.1.7 Angular rate measurement

a. A sensor with angular rate measurement capability shall have the

followingminimumoutputs:

1. the instantaneous angular rates

around the Boresight Reference

Frame(BRF)axesrelativetoinertialspace;

2. theMeasurementdate.

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b. The date of measurement shall be expressed as a (scalar) number

indicating the delay) relative to a known external time reference agreed

withthecustomer.

NOTE The intended use of this capability is either when

theattitudecannotbedeterminedortoprovidean

angularrate.

4.1.8 (Partial) image download

4.1.8.1 Image download

a. A sensor with the (partial) image download capability shall have the

followingminimumoutputs:

1. thesignalvalueforeachrelevantdetectorelement;

2. theMeasurementdate.

b. Any use of image compression (e.g. for transmission) shall be

documented.

NOTE The definition of the capability is intended to

exclude

‘lossy’ image compression, though such

compression can be a useful option under certain

circumstances.

4.1.8.2 Image Output Time

a. Thesuppliershallspecifythenumberofbitsperpixelusedtoencodethe

detectorimage.

b. The image output time shall be verified by test using the hardware

agreedbetweenthecustomerandsupplier.

NOTE1 The hardware used to perform the test is the

hardware used to download

 the image from the

starsensor.

NOTE2 Forexample:

• “TheStarSensorshallbecapableofperforming

a full Image Download of the entire Field of

View at 12‐bit resolution. The image output

timeshallbelessthan10seconds.”

• “TheStarSensorshallbecapable

ofperforming

a partial Image Download at 12‐bit resolution

ofan×nsectionoftheFieldofView.Theimage

outputtimeshallbelessthan10seconds.”

4.1.9 Sun survivability

a. Asensorwiththesunsurvivabilitycapabilityshallwithstanddirectsun

illuminationalongtheboresightaxis,foratleasta given periodof time

agreedwiththecustomer,withoutsubsequentpermanentdamage.

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b. Asensorwiththesunsurvivabilitycapabilityshallrecoveritsfullquoted

performancesafterthesunaspectanglehasbecomegreaterthanthesun

exclusionangle.

4.2 Types of star sensors

4.2.1 Overview

Thissubclausespecifiesthenomenclatureusedtodescribethedifferenttypesof

starsensors.Theirclassificationisbasedontheminimumcapabilitiestobemet

byeachtype.

The term star sensor is used to refer generically to any sensor using star

measurementstodriveitsoutput.Itdoesnotimply

anyparticularcapabilities.

NOTE The term Star Scanner is used to refer to a Star

Sensoremployedonspinningspacecraft.Thiskind

of sensor performs star measurements at high

angular rate (tens of deg/s). Formal capability

definition of the Star Scanner, together with

definedperformancemetricsareoutsidethescope



ofthisspecification.

4.2.2 Star camera

a. Astarcamerashallincludecartographyasaminimumcapability.

4.2.3 Star tracker

a. Astartrackershallincludethefollowingminimumcapabilities:

1. cartography;

2. startracking.

NOTE If the autonomous star tracking capability is

present, the cartography capability is internal to

the unit when initializing the tracked stars and

hencetransparenttotheground.

4.2.4 Autonomous star tracker

a. An autonomous star tracker shall include the following minimum

capabilities:

1. autonomousattitudedetermination(‘lostinspace’solution);

2. autonomousattitudetracking(withinternalinitialization).

b. The supplier shall document whether the autonomous attitude

determination capability is repetitively used to achieve the autonomous

attitudetracking.

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4.3 Reference frames

4.3.1 Overview

Thestandardreferenceframesaredefinedin3.2.3.

Otherintermediatereferenceframesaredefinedbythemanufacturersinorder

to define specific error contributions, but are not defined here,as they are not

usedintheformulationoftheperformancemetrics.Seealso

AnnexF.

4.3.2 Provisions

a. Any use of an IRF shall be accompanied by the definition of the IRF

frame.

b. Anyuseofanattitudequaternionshallbeaccompaniedbythedefinition

oftheattitudequaternion.

4.4 On-board star catalogue

a. The supplier shall state the process used to populate the on‐board star

catalogueandtovalidateit.

b. Theprocessstatedin

4.4ashallbedetailedtoalevelagreedbetweenthe

customerandthesupplier.

c. The supplier and customer shall agree on the epoch at which the on‐

boardstarcatalogueisvalid.

NOTE In this context, ‘valid’ means that the accuracy of

the on‐board catalogue is best (e.g.

the effect of

propermotionandparallaxisminimized).

d. The supplier shall state the epoch range over which performances are

metwiththeon‐boardstarcatalogue.

e. The supplier shall deliver the on‐board star catalogue, including the

spectralresponsesoftheopticalchainanddetector.

f. If the

 star sensor has the capability of autonomous attitude

determination, the supplier shall deliver the on‐board star pattern

catalogue.

g. The maintenance process of the on‐board star catalogue shall be agreed

betweenthecustomerandthesupplier.

NOTE1 Themaintenanceprocessincludesthecorrectionof

parallax and the

correction of the star proper

motionsintheon‐boardstarcatalogue.

NOTE2 Themaintenanceprocessincludesthecorrectionof

the on‐board catalogue errors identified in flight

(e.g.magnitude,coordinates).

h. The supplier shall state any operational limitations in the unit

performancecausedbytheon‐boardcatalogue(e.g.

autonomousattitude

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determinationnotpossibleforsomeregionsinthesky).Theselimitations

shallbeagreeduponbetweenthesupplierandthecustomer.

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Performance requirements

5.1 Use of the statistical ensemble

5.1.1 Overview

Performances have a statistical nature, because they vary with time and from

one realization of a sensor to another. This clause presents the knowledge

requiredtobuildperformancesup.Fulldetailscanbefoundin

AnnexE.

Onlyanenvelopeoftheactualperformancescanbeprovided.Centraltothisis

the concept of a

 ‘statistical ensemble’, made of ‘statistical’ sensors (i.e. not

necessarily built, but representative of manufacturing process variations) and

observations(dependingontimeandmeasurementconditions).

Three approaches (calledstatistical interpretations) can be taken to handle the

statistical

ensemble:

• Temporalapproach:performancesareestablishedwithrespecttotime.

• Ensemble approach: performances are established on statistical

 sensors

(i.e.notnecessarilybuilt),attheworstcasetime.

• Mixedapproach,whichcombinesboththeapproachesabove.

The conditions elected to populate the statistical

 ensemble are defined on a

case‐by‐case basis for each performance parameter, as described in the

followingclauses.

5.1.2 Provisions

a. Theperformancesshallbeassessedbyusingtheworst‐casesensorofthe

statistical

ensemble.

b. The statistical ensemble shall be characterized and agreed with the

customer.

c. The performances shall be assessed by using the sensor EOL conditions

agreedwiththecustomer.

NOTE The EOL conditions include e.g. aging effects,

radiationdose.

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5.2 Use of simulations in verification methods

5.2.1 Overview

Simulations efficiently support the verification of performances. A set of

simulationsprovidesan estimateofaperformance,obtainedbyprocessingthe

simulation results in a statistical fashion. Because the set of simulations is

limited, the performance estimated by simulations has a given accuracy,

essentiallydependingonthenumberofsimulations.

5.2.2 Provisions for single star performances

a. Software models of singlestar measurement error shall be validated for

singlestarperformance(atzerobodyrates)againston‐groundtestsusing

artificialstellarsources.

NOTE Denoting the confidencelevel to be verified asP

C,

and assuming that the performance confidence

level result to be obtained is to an accuracy ΔP 

with95%estimationconfidencelevel,thenumber

of Monte‐Carlo runs to be performed is greater

than

)1(4

−

.

5.2.3 Provisions for quaternion performances

a. Software models of attitude quaternion error shall be validated against

on‐ground tests using artificial stellar sources or with on ground tests

agreedbythecustomer.

NOTE1 Denoting the confidencelevel to be verified asP

C,

and assuming that the confidence result to be

obtained is to an accuracy ΔP with 95%

confidence,the number ofMonte‐Carloruns to be

performedisgreaterthan

)1(4

−

.

NOTE2 RefertoAnnex

E.1forfurtherdetails.

5.3 Confidence level

a. The following confidence level shall be agreed with the customer (see

5.5):

1. forthethermoelasticerror;

2. fortheFOVspatialerror;

3. forthepixelspatialerror;

4. forthetemporalnoise;

5. forthemeasurementdateerror;

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6. RefertoAnnexEforfurtherdetails.

NOTE1 A performance confidence level of 95% is

equivalent to a 2 sigma confidence level for a

Gaussiandistribution.

NOTE2 A performance confidence level of 99,7% used is

equivalent to a 3 sigma confidence level for a

Gaussiandistribution.

5.4 General performance conditions

a. The performance conditionsof the ‘statistical ensemble’shallbe used to

encompassthefollowingconditionsforEOL:

1. worst‐casebaseplatetemperaturewithinspecifiedrange;

2. worst‐caseradiationfluxwithinspecifiedrange;

3. worst‐case stray light from solar, lunar, Earth, planetary or other

sources.

NOTE1 InadditionvaluesforBOLcanbegiven.

NOTE2 Worst‐casestraylight

conditionsarewiththeSun,

Earth and (where appropriate) Moon

simultaneously at their exclusion angles together

with worst‐case conditions for any other light

sources.

b. Themaximummagnitudeofbodyrateshallbeused.

NOTE The maximum body rate is the worst case

conditionformostmissions.Forspecific

cases,the

worstcasecanbeadapted,e.g.toincludejitter.

c. The supplier shall identifytheworst case projection in BRF of the value

definedin

5.4d.

NOTE Different angular rates can be specified with

associatedrequiredperformance.

d. Themaximummagnitudeofangularaccelerationshallbeused.

NOTE The maximum angular acceleration is the worst

case condition for most missions. For specific

cases, the worst case can be adapted, e.g. to

includejitter.

e. The

 supplier shall identifytheworstcase projection inBRF of thevalue

de‐finedinbullet

d.

NOTE Different angular accelerations can be specified

withassociatedrequiredperformance.

f. Formultipleheadconfigurationtheworstcaseconditionsofangularrate

and stray light of each optical head shall be discussed and agreed

betweensupplierandcustomer.

g. Single star position measurement performance within the verification

simulations

shallbe:

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1. validated against on‐ground test data for fixed pointing

conditions,and

2. abletopredictmetricperformanceundertheseconditionswithan

accuracyof10%.

h. If test data is available for the individual error sources, the simulation

shallbevalidatedagainstthisdatawithanaccuracyof10%.



i. Detector error sources in the simulation shall also be validated using

directdatainjectionintotheelectronicsandanalysisofthetestoutputs.

j. The simulation allows the verification to cover the full range of

conditions, including stray light, finite rates/accelerations, full range of

instrumentmagnitudes,andtheworst‐

caseradiationexposure.

k. EOL simulations used to predict EOL performance shall be verified by

testcasesverifiableagainstmeasurableBOLdata.

l. Theimpactofindividualstarerrorsonthe overallrateaccuracyshallbe

providedviasimulation.

m. Noaidedtrackingshallbeconsidered.

5.5 General performance metrics

5.5.1 Overview

Clause 5.5 presents the general performance metrics for the error contributing

tothestarsensorperformances.In

AnnexH,anexampleofdatasheetbuilton

theperformancemetricsisgiven.

5.5.2 Bias

5.5.2.1 General

a. Theconfidencelevelspecifiedinclause5.3shallbeused.

b. The‘Ensemble’interpretationshallbeusedasfollows:

NOTE TheEnsembleinterpretationisasfollows:

• A

 statistical collection of sensors is arbitrarily

chosen.

• A given set of observations is arbitrarily

chosen.

• The specification for this type of variability is

‘lessthanthelevelSinconfidenceleveln%ofa



statistical ensemble of sensors/observations for

theworst‐casetime’.

c. The bias performance shall be specified for a defined ambient

temperature.

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NOTE The initial alignment is an instantaneous

measurement error at the time of calibration. For

the purposes of error budgeting it can be

consideredtobeaninvarianterror.

5.5.2.2 Contributing error sources

a. Thefollowingtypesoferrorsourceshallbeincluded:

1. On‐ground calibration error between the sensor Alignment

ReferenceFrame(ARF)andthe sensorBoresightReferenceFrame

(BRF).

NOTE This arises typically from accuracy limitations

within the measurement apparatus used to

performthecalibration.

2. Launchinducedmisalignmentsof

BRFwithrespecttoMRF.

3. Spatialerrorincaseofinertialpointing.

NOTE Refer to the

Annex G for the contributing error

sourcesdescription.

5.5.2.3 Verification methods

a. Thecalibrationshallbeperformedviaground‐basedtestusinganoptical

bench set‐up to determine the sensor Alignment Reference Frame

(ARF)‐sensorBoresightReferenceFrame(BRF)alignment.

b. The bias error shall be validated by analysis, test or simulation, taking

intoaccountcalibrationtestbenchaccuracy.

NOTE1

 Initial alignment verification cannot be done

without verification of the measurement accuracy

oftheset‐upusedforcalibration.

NOTE2 E.g. “The Star Sensor initial alignment shall have

an initial alignment error (X‐, Y‐axes rotation) of

less than 10 arcsec at a quoted ambient

temperature(thetemperature

duringalignment).”

5.5.3 Thermo elastic error

5.5.3.1 General

a. Theconfidencelevelspecifiedinclause5.3shallbeused.

b. The‘Ensemble’interpretationshallbeused(seeNOTEin

5.5.2.1b).

NOTE The ‘Ensemble’ interpretation is selected here as

thetimevariationoftheseerrorsisslow–theyare

to all intents and purposes biases for practical

measurementscenarios.

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5.5.3.2 Contributing error sources

a. Error sources that gradually change the alignment of the sensor

Mechanical Reference Frame (MRF) and the sensor Boresight Reference

Frame(BRF)fromthestartofthein‐flightmissionshallbeincluded.

NOTE E.g.“Thethermalsensitivitytotemperatureofline

ofsightstabilityshallbelessthan1

arcsec/Kelvin.”

5.5.3.3 Verification methods

a. Thermallyinducederrorcontributionstothethermoelasticerrorshallbe

verified by the use of thermal models supported and validated by

groundtestresultsperformedunderthermalvacuumconditions.

5.5.4 FOV spatial error

5.5.4.1 General

a. Theconfidencelevelspecifiedinclause5.3shallbeused.

b. The‘Ensemble’interpretationshallbeused(seeNOTEin

5.5.2.1b).

c. The performance shall be specified under the related performance

generalconditions.

5.5.4.2 Contributing error sources

a. ContributingErrorSourcesshallinclude:

1. pointspreadfunctionvariabilityacrosstheFOV;

2. residual of calibration of focal length (including its temperature

sensibility)andopticaldistortions(includingchromatism);

3. residual of aberration of light in case where it is corrected at

quaternionlevelandnotatstarlevel;

CCD,CTEeffect(includingitsdegradationsduetoradiations);

5. catalogueerror(includingstarpropermotionandparallax).

5.5.4.3 Verification methods

a. ThemeasurementoftheFOVspatialerrorshallbeperformedviaground 

test (for contributing error sources

5.5.4.2a.1 and 5.5.4.2a.2) and by

analysis (for contributing error sources

5.5.4.2a.3, 5.5.4.2a.4 and

5.5.4.2a.5).

b. Radiationeffectsshallbesupportedbytestresults.

NOTE E.g. “The Star Sensor shall have a  FOV spatial

errorlessthan10arcsecon X,Yaxesand40arcsec

on Z axis for spatial period smaller than

5degrees.”

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5.5.5 Pixel spatial error

5.5.5.1 General

a. Theconfidencelevelspecifiedinclause5.3shallbeused.

b. The‘Ensemble’interpretationshallbeused.

c. The performance shall be specified under the related performance

generalconditions.

5.5.5.2 Contributing error sources

a. Contributingerrorsourcesshallconsistofatleast:

1. detectorPhotoResponseNonUniformity(PRNU);

2. detectorDarkSignalNonUniformity(DSNU);

3. detectordarkcurrentspikes‐ifrelevantaccording tothedetector

technology;

4. detector Fixed Pattern Noise (FPN)‐if relevant according to the

detectortechnology;

5. starcentroid

computationerror(interpolationerror).

b. Allothererrorsourceswithrelevantspatialbehaviourshallbeidentified

bythesupplierandusedfortheassessmentofperformances.

5.5.5.3 Verification methods

a. Contributingerrorsourcesshallbeverifiedbyongroundtests.

b. Pixel spatial errors shall be verified by analysis and simulations using

verifiedbudgetsofcontributingerrorsourcesmethods.

NOTE E.g. “The Star Sensor shall have a pixel spatial

errorof less than 5arcseconds(resp.30)aroundX

and Y axes (resp. Z axis) for spatial period of 400

arcsecond, and less than 2 arcseconds (resp 10)

around X and Y axes (resp. Z axis) for spatial

periodof100arcsecond.”

5.5.6 Temporal noise

5.5.6.1 General

a. Theconfidencelevelspecifiedinclause5.3shallbeused.

b. The‘temporal’interpretationshallbeused,andtheperformanceshallbe

specifiedundertherelatedperformancegeneralconditions.

5.5.6.2 Contributing error sources

a. TheContributingErrorSourcesshallinclude:

1. shotnoiseonstarsignal;

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2. shotnoiseonbackgroundsignalanddarkcurrent;

3. read‐outnoise;

4. quantificationnoise;

5. datationnoise.

NOTE1 Temporal noise depends on exposure time and

detectortemperatures.

NOTE2 Noise contributors at star level depend on star

magnitude angular rates/acceleration, and

optics/detector characteristics (e.g. exposure time,



opticalcontamination,transmissionloss,defocus).

NOTE3 Datation noise is the temporal noise part of the

measurementdateerrordescribedin

5.5.8.

5.5.6.3 Verification Methods

a. Temporalnoiseshallbeestimatedbysimulation.

b. Errorsourcecontributors

5.5.6.2a.1,5.5.6.2a.2,5.5.6.2a.3,5.5.6.2a.4shallbe

validatedagainstongroundtestdataat(BOL)forfinitescenarios.

c. Error source contributor

5.5.6.2a.5 (datation noise) shall be assessed by

analysis.

NOTE1 Night Sky tests are not used as single verification

methodduetoexperimentalconditions.Nightsky

tests can be used to assess temporal noise in

addition to other required verification methods

(simulationsandon‐groundtests).

NOTE2 E.g.“The Star

 Sensor shall have  temporal noise of

less than 10arcsecaround any axisup to 10 deg/s

atEOLandforaccelerationsupto1,0deg/s².”

5.5.7 Aberration of light

5.5.7.1 General

a. The supplier shall document what type of relativistic correction is

performed.

b. The supplier shall document the maximum error and minimum

frequencyofthespacecraftvelocityprovidedtothesensor.

5.5.7.2 Contributing error sources

a. ThecontributingErrorSourcesshallinclude:

1. Absolutelinearvelocityofthespacecraftwithrespecttothesun.

2. Accuracy of the velocity information (or propagation) used for

correction.

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5.5.7.3 Verification methods

a. The correction software shall be validated, comparing the computed

correctiontermwiththeanalyticalexpression.

NOTE1 Thiserrorcorrectionisdifficulttoverifysinceitis

atheoreticaltermoferror.

NOTE2 E.g. “The relativistic effect has an impact of less

than 0,07’’ (3σ) at quaternion level.

 The needed

accuracyofthevelocityofthespacecraftdelivered

to the star sensor shall be better that 100m/s, at a

frequencyof0,1Hz.”

5.5.8 Measurement date error

a. Theconfidencelevelspecifiedinclause5.3shallbeused.

b. TheMeasurementdateErrorshallbeverifiedbytest.

NOTE E.g. “The Measurement date Error shall be less

than0,1ms.”

5.5.9 Measured output bandwidth

a. The bandwidth shall be verified by analysis of the Integration Time,

output

Sampling Time and any on‐board data filtering that can be

present.

b. On‐groundtestsmaybeperformed.

NOTE E.g. “The

Star Sensor shall have a Measured

OutputBandwidthofgreaterthan10Hz.”

5.6 Cartography

a. For star position measurements, the performance conditions of the

‘statistical

ensemble’shallbeusedtoencompassthefollowingconditions

forBOL:

1. worst‐casestarlocationinFOV;

2. worst‐caseStarMagnitudewithinspecifiedrange.

5.7 Star tracking

5.7.1 Additional performance conditions

a. For star position measurements, the performance conditions of the

‘statistical

ensemble’shallbeusedtoencompassthefollowingconditions

forBOL:

1. worst‐casestarlocationinFOV;

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2. worst‐caseStarMagnitudewithinspecifiedrange.

5.7.2 Single star tracking maintenance probability

a. Thefollowingconditionsshallbemet:

1. quote the maximum body rate ω

CROSS, MAX around the sensor

Boresight Reference Frame (BRF) X‐ or Y‐axes and ωZ, MAX

around the BRF Z‐axis for which the single star tracking

maintenanceprobabilityisachievedoverthedefinedtimeperiod;

2. quote the maximum body angular acceleration around the sensor

boresight reference frame (BRF) X‐ or

 Y‐ axes and the maximum

body angular acceleration around the BRF Z‐axis for which the

single star tracking maintenance probability is achieved over the

definedtimeperiod.

NOTE E.g. “The Track Maintenance Probability shall be

greater than 99% over a time period of 1  minute

for a tracked

 Star Image (of magnitude less than

tbdmi)remainingwithinthesensorFOV,forrates

aroundanyaxisofupto100arcsec/satEOL,with

accelerationsupto10arcsec/s².”

5.8 Autonomous star tracking

5.8.1 Additional performance conditions

a. For star position measurements, the performance conditions of the

‘statistical

ensemble’shallbeusedtoencompassthefollowingconditions

forBOL:

1. worst‐casestarlocationinFOV;

2. worst‐caseStarMagnitudewithinspecifiedrange.

b. The followingadditionalperformance metrics shallbe established:track

maintenanceprobability,asspecifiedin

5.7.2.

c. Forthestatistical

ensemble,provisionsin5.2.2shallbeapplied.

NOTE The same definition for the ‘statistical ensemble’

givenin5.1.1applies.

5.8.2 Multiple star tracking maintenance level

a. Thefollowingconditionsshallbemet:

1. quote the maximum body rate ω

CROSS, MAX around the sensor

Boresight Reference Frame (BRF) X‐ or Y‐axes and ωZ, MAX

around the BRF Z‐axis for which the multiple star tracking

maintenancelevelisachievedoverthedefinedtimeperiod;

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2. quote the maximum body angular acceleration around the sensor

boresight reference frame (BRF) X‐ or Y‐ axes and the maximum

body angular acceleration around the BRF Z‐axis for which the

single star tracking maintenance probability is achieved over the

definedtimeperiod;

3. Thegeneralprovisionsin

5.2.2shallbeapplied.

NOTE E.g.“TheMaintenanceLevelofStarTracksshallbe

atleast5tra cksforatotaltimeof995swithinany

1000s period, for rates around any axis of up to

100arcsec/s at EOL, andforaccelerationsupto10

arcsec/s².”

5.9 Autonomous attitude determination

5.9.1 General

a. When Autonomous Attitude Tracking is performed by using repetitive

AutonomousAttitudeDeterminationthemetricsrelativetoautonomous

attitudetrackingspecifiedin

5.10shallbeapplied.

NOTE Thiscapabilityis often referred toastheabilityto

solve the ‘lost in space’ problem. The orientation,

orattitude, measurement is nominally in the form

of a quaternion that parameterizes the

transformation between the Inertial reference

frameandthesensor‐definedreferenceframe.The

determination is nominally performed by

comparing star images measured on a detector to

knownstarpositionsandcharacteristicsstoredina

starcataloguewithinthesensor.

b. When Autonomousattitude determinationisonly used for autonomous

attitude tracking initializationthe general performance metrics shall not

beused.

5.9.2 Additional performance conditions

5.9.2.1 Autonomous attitude determination

a. The Autonomous attitude determination shall be subjected to the

followingattitudedeterminationprobabilityperformancemetrics:

1. probabilityofcorrectattitudedetermination;

2. probabilityoffalseattitudedetermination;

3. probabilityofinvalidattitudedetermination.

NOTE Thevalidityflagneedsnotaperformancemetric.

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5.9.2.2 Lunar and planetary effects on performance

a. If a statement of operation with the Moon in the FOV is specified, the

attitude determination probabilities shall be quoted for the ‘Moon in

FOV’scenario.

b. IfastatementofoperationwithplanetaryobjectsintheFOVisspecified,

theattitudedeterminationprobabilitiesshallbequotedforthe‘Planet

in

FOV’scenario.

c. The attitude determination probabilities specification shall be quoted

with the maximum number of False Stars in the FOV for which the

specificationissatisfied.

5.9.3 Verification methods

a. Theprobabilitiesofattitude determination specificationshallbeverified

byapplyingthegeneralprovisionsin

5.2.2and5.2.3.

b. Functionalverificationmaybeperformedbymeansofanightskytest.

5.9.4 Attitude determination probability

a. ProbabilityofCorrectAttitudeDetermination:

1. Thecorrectattitudethresholdshallbespecified.

NOTE E.g. “The correct attitude threshold shall be

0,1degree around X an Y axis and 0,3 degree

aroundZaxis”

2. Theprobabilityofcorrectattitudedeterminationshallbeestimated

consideringallpossibleinitialpointing

directionswithinadefined

regionwithinthecelestialsphere.

3. Theprobabilityofcorrectattitudedeterminationshallbeestimated

undertheconditionsgivenin

5.4and5.9.2.

4. The probability of correct attitude determination shall be verified

usingthemethodspecifiedin

5.9.3.

NOTE E.g. “An example of requirement specification is

the following: the probability of correct attitude

determination within 10s shall be greater than

99,99% for random initial pointings within the

entirecelestialsphere,forratesaroundanyaxisof

upto 100arcsec/s at EOL and for accelerations

up

to10arcsec/s².”

b. ProbabilityofFalseAttitudeDetermination:

1. The probability of false attitude determination shall be estimated

consideringallpossibleinitialpointingdirectionswithinadefined

regionwithinthecelestialsphere.

2. The probability of false attitude determination shall be estimated

undertheconditionsgivenin

5.4and5.9.2.

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3. The probability of false attitude determination shall be verified

usingthemethodspecifiedin

5.9.3.

NOTE E.g. “The probability of false attitude

determination within 10s shall be less than 0,1%

for random initial pointings within the entire

celestialsphere, for ratesaroundanyaxisofup to

100arcsec/s at EOL and for accelerations up to

10arcsec/s².”

c. ProbabilityofInvalid

AttitudeSolution:

1. The probability of invalid attitude solution shall be estimated

consideringallpossibleinitialpointingdirectionswithinadefined

regionwithinthecelestialsphere.

2. The probability of invalid attitude determination shall be

estimatedundertheconditionsgiveningivenin

5.4and5.9.2.

3. The probability of invalid attitude determination shall be verified

usingthemethodspecifiedin

5.9.3.

NOTE E.g. “The probability of invalid attitude solution

shallbelessthan0,1%forrandominitialpointing

within the entire celestial sphere, for rates around

any axis of up to 100arcsec/s at EOL and for

accelerationsupto10arcsec/s².”

5.10 Autonomous attitude tracking

5.10.1 Additional performance conditions

a. ForbothBOLandEOL,theperformancemetricsshallbespecifiedeither:

⎯ Fromthewholecelestialsphereincludingthevaultinthestatistics,

or

NOTE The statistical

 ensemble is then composed of

measurements randomly performed  on the entire

celestialvault.

⎯ Fromasetoffixeddirectionsinthecelestialsphere.

b. If the metrics are specified from a set of fixed directions in the celestial

sphere when satisfying conditions detailed in

a the following shall be

met:

1. assess the metrics for each direction, limiting the statistical



ensembletomeasurementsperformedinthisdirectiontocompute

theperformance;

2. Specifyallorpartofthefollowing:

(a) Themeanperformanceamongallperformancesachievedin

thedirectionsofthecelestialsphere,

(b) Thevalueachievedonn%ofthecelestialsphere,

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NOTE This is the performance achieved for n% of the

pointing directions within the whole celestial

vault. If n is not quoted, a value of 99% is

assumed.

celestialsphere.

NOTE Thisdirectionisrelatedtotheworstdistribution

of

starsoverthestarsensorFieldofView,takinginto

accountembeddedalgorithmsandcatalogues.The

statistical

 ensemble is then reduced to

measurementsperformedinthisdirection.

c. Performances may also be specified for a restricted area of the celestial

sphereagreedwiththecustomer,inwhichcasetheperformancemetrics

arethenspecifiedinthesameway,limitingthestatistical

ensembletothe

specifiedarea.

d. ForLunarandplanetaryeffectsonperformancethefollowingconditions

shallbemet:

1. IfastatementofoperationwiththeMoonintheFOV isspecified,

quote the probability ofmaintenance of tracking for the ‘Moon in

FOV’scenario.

2. If a statement of operation with planetary

 objects in the FOV is

specified, quote the probability of maintenance of tracking the

‘PlanetinFOV’scenario.

e. FortheeffectofFalseStarsthefollowingconditionshallbemet:

Quote the maintenance level of tracking with the maximum number of

FalseStarsintheFOVforwhichthe

specificationisapplicable.

f. Forthe effectofsingleeventupsets(SET’s)thefollowingconditionshall

bemet:

Quote the maintenance level of tracking with the maximum number of

SET’spersecondforwhichthespecificationisapplicable.

5.10.2 Maintenance level of attitude tracking

5.10.2.1 General

a. The performance shall be specified under the conditions given in 5.10.1

and

5.10.2.2a.

5.10.2.2 Verification methods

a. The maintenance level of tracking shall be verified by applying the

generalprovisionsin

5.2.2.

NOTE E.g. “The maintenance level of tracking shall be

more than 995s within a 1000s period, for rates

around any axis of up to 100arcsec/s at EOL, and

foraccelerationsupto10arcsec/s².”

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5.10.3 Sensor settling time

a. Theperformanceshallbespecifiedundertheconditionsgivenin5.10.1.

b. Forlunarandplanetaryeffectson performancethe following conditions

shallbemet:

1. IfastatementofoperationwiththeMoonintheFOV isspecified,

quotetheSensorSettlingTimeforthe‘MooninFOV’scenario.

2. If a statement of operation with planetary objects in the

 FOV is

specified, quote the Sensor Settling Time for the ‘Planet in FOV’

scenario.

c. For the effect of False Stars the following condition shall be met: Quote

theSensorSettlingTimewiththemaximumnumberofFalseStarsinthe

FOVforwhichthespecificationisapplied.

NOTE The

 effect of convergence of internal algorithm

shallbeconsidered

d. The Sensor Settling Time shall be verified by applying the general

provisionsin

5.2.2.

NOTE E.g.“SensorSettlingTime shallbe less than 5s for

more than 99% of random initial pointing within

the entire celestial sphere, for rates around any

axis of up to 100arcsec/s at EOL and for

accelerationsupto10arcsec/s².”

5.11 Angular rate measurement

5.11.1 Additional performance conditions

a. Additionalperformanceconditions,definedin5.10.1shallbeapplied.

b. Contributingerrorsourcesshallbeestablished.

NOTE They are a function of the precise technique used

todeterminetherate.

5.11.2 Verification methods

a. Performanceat finiteratesandaccelerations,andforall scenariosunder

thespecifiedconditions,shallbeverifiedbysimulation.

NOTE E.g. The Star Sensor shall have an angular rate

measurement around any BRF axis of less than

100arcsec/s, at rates around any axis of up to

10deg/s

at EOL and for accelerations up to

1deg/s².

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5.12 Mathematical model

a. The supplier shall deliver a temporal functional mathematical model of

theperformanceofthestarsensor.

NOTE This is essential for some capabilities

(e.g.autonomousattitudetracking).

b. The functional mathematicalmodelshall be representative of the sensor

actualtemporalperformancesforrealistickinematicprofiles.

c. The functional mathematical

model shall include environmental

parameters.

d. The functional mathematical model shall be established with customer

approvedmethods.

e. The functional mathematicalmodel shallbe validated againstthe actual

temporalperformancesofthesensor.



f. Thesuppliershalldeliver:

⎯ eithertheFMMsoftwareusedbythesuppliertoassessthesensor

performancesanditsassociateddocumentation(e.g.usermanual)

inaformatagreedwiththecustomer,or

⎯ the FMM DRD ofthesensormodelused by thesupplierto assess

thesensorperformances,inconformancewith

AnnexA.

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Annex A (normative)

Functional mathematical model (FMM)

description - DRD

A.1 DRD identification

A.1.1 Requirement identification and source

document

ThisDRDiscalledfromECSS‐E‐ST‐60‐20,requirement5.12f.

A.1.2 Purpose and objective

The functional mathematical models are established to serve as input for

detailedAOCSanalysesanddetailedperformancesimulations.

A.2 Expected response

A.2.1 Scope and content

<1> Introduction

a. The FMM description shall contain a description of the purpose,

objective,contentandthereasonpromptingitspreparation.

b. Any open issue, assumption and constraint relevant to this document

shallbestatedanddescribed.

c. Statusandlimitationsofthemodelshallbedescribedindetail.

<2> Applicableandreferencedocuments

a. TheFMMdescriptionshalllisttheapplicableandreferencedocumentsin

supporttothegenerationofthedocument.

<3> Definitionsandabbreviations

a. The FMM description shall list the applicable directory or glossary and

themeaningofspecifictermsorabbreviationsutilizedintheFMM.

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<4> Functionalmathematicalmodel(FMM)

a. The steps from the actual quaternion in inertial frame to the sensor

outputsshallbedocumented,including:

1. staridentification;

2. patternrecognition;

3. star corrections (e.g. optical aberration correction, relativistic

aberrationcorrection);

4. quaternioncomputation;

5. filtering.

b. TheoutputsoftheFMMshallinclude:

1. themeasuredquaternion

andtimedeliveredbythesensor;

2. thestarmeasurementsandtimesdeliveredbythesensor;

3. thestaridentificationinformation.

c. TheoutputsoftheFMMshall includethe outputs of the sensor detailed

inclause

4(see4.1.2.2,4.1.3.2,4.1.4.2,4.1.5.2,4.1.6.2,or4.1.8.2),according

tothesensorcapabilities.

d. TheparametersoftheFMMshallbedocumented.

e. Modelling constraints and critical implementation issues shall be

describedandtheirrelevanceonperformanceshallbeindicated.

f. The FMM shall present the expected temporal outputs of the sensor

modelforgiveninputprofiles.

<5> Modes

a. For sensors with the autonomous attitude determination capability, the

FMM description shall include the autonomous attitude determination

capability.

b. For sensors with the autonomous attitude tracking capability, the FMM

descriptionshallincludetheautonomousattitudetrackingcapability.

<6> Softwaretools

a. The software tools to be used for development of the FMM shall be

specified.

<7> Filesandlists

a. Thefollowinginformationshallbeattachedtothedocument:

1. identificationofdeliveredcomputerfiles;

2. FMMsourcelistsbasedonappliedtools.

A.2.2 Special remarks

None.

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Annex B (informative)

Ancillary terms in Star Sensors

B.1 Overview

This annex standardizes the meaning of terms that, although not used in this

document, are used in star sensors engineering. It also presents the

measurementerrormetrics.

B.2 Time and frequency

B.2.1 frame frequency

inverseoftheframetime

B.2.2 frame time

time interval between two consecutive beginnings of integration time of each

outputofasingleOpticalHead

B.2.3 internal sampling time

time interval between the Measurement Dates of consecutive measurements

fromasingleOpticalHead

B.2.4 internal sampling frequency

inverseoftheinternalSamplingTime

B.2.5 latency

timebetweenthemeasurementdateandtheoutputdate

B.2.6 output date

dateofthefirstavailabilityoftheoutputdataforuseexternaltothesensor

NOTE Sensors can either be operated asynchronously

(output provided when available based on sensor

clock)orsynchronously(whenthesensorisaslave

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to an external clock pulse). In the latter case the

output data sometimes cannot be accessed and

placed in TM until some time after it was made

available. This additional delay is specifically

excludedfromthelatencydefinition.

B.2.7 output rate

rate at which the sensor  delivers its data for each output of a single Optical

Head

B.3 Angles of celestial bodies

B.3.1 acquisition angle with Moon angle (AAM)

lowest Aspect Angle of the Full Moon at which the Autonomous Attitude

Determinationisoperatingsuccessfullybutwithdegradedperformance

NOTE1 AAMislessorequaltoMEAandisexpectedtobe

greaterorequaltoTAM.

NOTE2 AAM and TAM define the robustness of the

behaviourof

thestarsensorwhentheMoonenters

thefieldofview.

B.3.2 tracking angle with moon in the FOV (TAM)

lowest Aspect Angle of the Full Moon at which the Autonomous Attitude

Trackingisstilloperatingsuccessfullybutwithdegradedperformance

NOTE1 TAMislessorequaltoMEA.

NOTE2 TAMandAAM(see

B.3.1)definetherobustnessof

the behaviour of the star sensor when the Moon

entersthefieldofview.

B.4 Full sky

celestial sphere covering the complete 4π steradian solid angle with respect to

thesensor

B.5 Measurement error metrics

B.5.1 Overview

Thisclause declinesthe measurement error metrics,prior to application to the

Star Sensor measurement error specification. A link to the nomenclature for

traditionalerrormetricsisalsoincludedtoaidmigrationtothenewmetricset.

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Annex F establishes the expression of the angular error on which the angular

metricsisapplied:

()

⎥

⎦

⎤

⎢

⎣

⎡



B.5.2 time interval for a metric

thetimeintervaltXforametricXisdefinedasatimeperiodwithstarttimetSX

andlengthτ

X

B.5.3 absolute measurement error (AME)

theabsolutemeasurementerror(AME(t))istheangularerror

()

atatimet:

()

ttAME

=)( 

NOTE Thisisillustratedschematicallyin

FigureB‐1fora

singleaxisrotationcase.

B.5.4 quaternion absolute measurement error (AMEq)

AMEinwhichtheangularerrorisderivedfromthemeasuredquaternion

meas

q 

NOTE The quaternion

meas

q  is used to build the frame

transform matrix

XRFIRF

−

 from an inertial

referenceframe(IRF)toasensor‐definedreference

frame (see clause

3.2.3), generically called XRF.

The error

()

⎥

⎦

⎤

⎢

⎣

⎡

 is then computed from

XRFIRF

−

accordingtoAnnexF.

B.5.5 star absolute measurement error (AMEs)

AME in which the angular error is derived from the sensor‐measured star

position

][

,, MEASYMEASXMEAS

sss = 

NOTE1 The sensor‐measured star position

][

,, MEASYMEASXMEAS

sss

 is defined as two

angular rotations parameterizing the

transformationbetweenasensordefinedreference

frame(heredenoted genericallyby ‘XRF’) and the

Stellar Reference Frame defined by the observed

star for the specific star. The X and Y rotations

provides the full parameterization since the third

rotation is zero by

definition. In this case, the star

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position measurement

MEAS

 is used to build the

frame transform matrix

XRFIRF

−

, from which the

error

()

⎥

⎦

⎤

⎢

⎣

⎡

canbecomputedaccordingto

AnnexF.

NOTE2 Theusualparameterizationistousethe‘2’and‘1’

Euler rotations (within the 3‐2‐1 convention‐the

anglesaresmallandsotheorderoftherotationsis

not important). Note that, in this definition, these

rotationerrorsare(inthesmallanglelimit)around



the X‐ and Y‐axes of the Stellar Reference Frame

(SRF), which are perpendicular to the LOS to the

starinthefieldofview.

B.5.6 absolute rate measurement error (ARME)

the difference between the measured and actual angular rate components,

relativetoitstargetframe,definedas:

()

BRF

tARME

ωω

−= 

where

BRF

 and

BRF

 are respectively the measured and actual angular rate

vectoraroundtheBoresightReferenceFrameaxes,relativetoinertialspace.

NOTE TheAbsoluteRateMeasurementErrorisspecified

for each axis by the absolute valueof the relevant

vectorcomponent.

B.5.7 mean measurement error (MME)

themeanvalue

oftheangularerror

(

)

overatimeinterval

:

()

=ΔtMME where

()

∫

dtt



NOTE Thisisillustratedschematicallyin

FigureB‐1fora

singleaxisrotationcase.

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)(tAME

FigureB‐1:AME,MMEschematicdefinition

B.5.8 quaternion mean measurement error (MMEq)

MME in which the angular error is derived from the measured quaternion

meas

q .

NOTE Seenotein

B.5.4.

B.5.9 star mean measurement error (MMEs)

MME in which the angular error is derived from the measured star position

MEAS

s 

NOTE Seenotein

B.5.5.

B.5.10 relative measurement error (RME)

therelativemeasurementError(RME(t))isdefinedasfollows:

()

−=,tRME



NOTE Thisisillustratedschematicallyinforasingleaxis

rotationcasein

FigureB‐2.

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FigureB‐2:RMESchematicDefinition

B.5.11 quaternion relative measurement error (RMEq)

RMEinwhichtheangularerrorisderivedfromthemeasuredquaternion

meas

q 

NOTE Seenotein

B.5.4.

B.5.12 star relative measurement error (RMEs)

RME in which the angular error is derived from the measured star position

MEAS

s 

NOTE Seenotein

B.5.5.

B.5.13 measurement drift error (MDE)

themeasurementdrifterror(MDE(t))isthedifferencebetween two successive

meanmeasurementerrors,separatedby

MDE

asfollows:

() ()

∫∫

++Δ+

Δ+

−=

ττ

dttdttMDE

MDE



where the lengths of the two successiveintervals are set to identical values

;

bothintervalsarecontainedinalongerintervalwithlength

OBS



NOTE Thisisillustratedschematicallyin

FigureB‐3fora

singleaxisrotationcase.

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MDE

OBS

FigureB‐3:MDESchematicDefinition

B.5.14 quaternion measurement drift error (MDEq)

MDE in which the angular error is derived from the measured quaternion

meas

q .

NOTE Seenotein

B.5.4.

B.5.15 star measurement drift error (MDEs)

MDE in which the angular error is derived from the measured star position

MEAS

s 

NOTE Seenotein

B.5.5.

B.5.16 rotational error

Each of the metrics defined in clause B.5 is parameterized by 3 rotations



around axis ‘j’ of a specific frame F. With respect to this frame, the rotational

errorR

jaroundeachaxis‘j’oftheFframeisgivenby:



NOTE1 Therotationalerrorisillustratedin

FigureB‐4.

NOTE2 The applicability of the specification formulation

intermsofdirectionalandrotationalerrorsallows

separate specificationandperformancestatements

relative to the direction of the sensor LOS and

around the sensor LOS. This is useful since the

performance in these 2 areas is typically

significantly different

 for single optical head

configuration and hence requires separate

specification.

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B.5.17 directional error

The directional error D(j) for axis ‘j’ is defined as the half‐cone angle between

the measuredand reference position of axis ‘j’ and is given (for small rotation

angles)by:

lkj

εε

+= 

where‘k’and‘l’arethetwoaxesperpendiculartoaxis‘j’.

NOTE1 Thedirectionalerrorisillustratedin

FigureB‐4.

NOTE2 The applicability of the specification formulation

intermsofdirectionalandrotationalerrorsallows

separate specificationandperformancestatements

relative to the direction of the sensor LOS and

around the sensor LOS. This is useful since the

performance in these 2 areas is typically

significantly different

 for single optical head

configuration and hence requires separate

specification.

Axis ‘j’

Axis ‘k’

Axis ‘l’

FigureB‐4:RotationalanddirectionalErrorGeometry

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B.6 Spatial errors

Some error contributors vary with theposition ofastar on the detector.These

errors(e.g.fieldofviewerrors,pixelerrors)canbetackledbyspatialerrors.

Themathematicalexpressionsarethesameastheonespresentedintheclauses

above,inwhichthetimetisessentiallyreplacedby

aspatialpositionx.

Foramoregeneraldomainvariable,x,theindicescanberedefinedasfollows:

() ()

xxAME



() ()

∫

xdxxMME



()() ()

∫

−=Ω

xdxxxRME

εε

 

x Ω∈ 

()

() ()

∫∫

ΩΩ

−

=ΩΩ

1,21

1,2,

2,1,

xdxxdxMDE

εε



Where



is a specified region of parameter space and

x Ω∈  means that x

lieswithinthatregion.

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Annex C (informative)

Optional features of star sensors

C.1 Overview

Thisannexdefinesoptionalfeaturesorcapabilitiesofstarsensors.Itfollowsthe

same structure as the clause

4 to allow for a direct link between requirements

andoptions.

C.2 Cartography

A sensor with cartography capability can have the following additional

outputs:measurementofstarmagnitudeofeachdetectedstarimage.

NOTE The star images obtained need not be captured at

thesameinstantintime.

C.3 Star tracking

a. Thefollowingadditionalinputstolaunchtrackingcanbeprovided:

1. theangularaccelerationandjerkofthesensorBRFwithrespectto

theIRF,withtheirvaliditydates;

2. theaccuracyofsuppliedinputs.

NOTE1 Angular acceleration and jerk are supplied in the

form of 3‐dimension vectors

 giving the angular

acceleration and jerk of the sensor BRF with

respect to the IRF. These vectors are expressed in

thesensorBRF.

NOTE2 In the case of external inputs coming from the

spacecraftthe starsensorsupplier canindicate the

minimum required accuracy for supplied data in

order

toproperlyswitchintotracking.

b. A sensor with the star tracking capability can have the following

additionaloutputs:measurementofstarmagnitudeforeachtrackedstar

image.

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C.4 Autonomous star tracking

a. Thefollowingadditionalinputstolaunchtrackingcanbeprovided:

1. TheangularaccelerationandjerkofthesensorBRFwithrespectto

theIRF,withtheirvaliditydates.

2. Theaccuracyofsuppliedinputs.

NOTE1 Angular acceleration and jerk are supplied in the

form of 3‐dimension vectors

 giving the angular

acceleration and jerk of the sensor BRF with

respect to the IRF. These vectors are expressed in

thesensorBRF.

NOTE2 InthecaseofexternalinputscomingfromtheS/C

thestarsensorsuppliercanindicatetheminimum

required accuracy for supplied data in order

to

properlyswitchintotracking.

b. A sensor with the autonomous star tracking capability can have the

following additional outputs: measurement of star magnitude for each

trackedstarimage.

C.5 Autonomous attitude determination

a. Asensorwithautonomousattitudedeterminationcanhavethefollowing

additionaloutputs:

1. ameasurementqualityindexorflag,estimatingtheaccuracyofthe

determinedattitude;

2. An inertial angular rate measurement projected on a  sensor‐

definedreferenceframe;

3. a list of the star catalogue numbers for each star

 used in the

determination;

4. the position of each star image with respect to a defined sensor

referenceframe;

5. measurementofstarmagnitudeforeachtrackedstarimage.

6. the identification of the optical head(s) used for the attitude

determinationwhenmultipleheadconfigurationisused.

C.6 Autonomous attitude tracking

a. Thefollowingadditionalinputstolaunchtrackingcanbeprovided:

1. theangularaccelerationandjerkofthesensorBRFwithrespectto

theIRF,withtheirvaliditydates;

2. the accuracy of supplied inputs including, in the case of attitude

control,theaccuracyaroundeachaxisofthesensor

BRF.

NOTE1 Angular acceleration and jerk are supplied in the

form of 3‐dimension vectors giving the angular

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acceleration and jerk of the sensor BRF with

respect to the IRF. These vectors are expressed in

thesensorBRF.

NOTE2 InthecaseofexternalinputscomingfromtheS/C

the star sensor supplier indicates the minimum

required accuracy for supplied data in order to

properlyswitchintotracking.



b. A sensor with autonomous attitude tracking capability can have the

followingadditionaloutputs:

1. ameasurementqualityindexorflagestimatingtheaccuracyofthe

determinedattitude;

2. an angular rate measurement around a sensor defined reference

frame;

3. a list of the star catalogue numbers for each star

used in the

determination;

4. thepositionofeachStarImagewithrespecttoadefinedreference

frame;

5. the identification of the optical head(s) used for the attitude

trackingwhenmultipleheadconfigurationisused.

C.7 Angular rate measurement

a. A sensor with angular rate measurement capability can have the

followingoutputs:

1. ameasurementqualityindexorflag,estimatingtheaccuracyofthe

determinedangularrate;

2. a validity index or flag, estimating the validity of the determined

angularrate.

C.8 Types of star sensors

C.8.1 Star camera

a. A star camera can includethe following additionalcapabilities:(partial)

imagedownload.

C.8.2 Star tracker

a. Astartrackercanincludethefollowingadditionalcapabilities:

1. autonomousstartracking;

2. (partial)imagedownload.

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C.8.3 Autonomous star tracker

a. An autonomous star tracker can include the following additional

capabilities:

1. cartography;

2. startracking;

3. autonomousstar tracking (attitude acquisition with assisted

attitudedetermination);

4. autonomousattitudetracking(withdirectinitialization);

5. angularratemeasurement;

6. (partial)imagedownload.

C.8.4 Summary

a. The specified minimum and additional capabilities for each type of

sensoraresummarizedin

TableC‐1.

TableC‐1:Minimumandoptionalcapabilitiesforstar

sensors

Capabilities

Typeofsensor



Cartography

StarTracking

AutonomousStarTracking

AutonomousAttitudeDetermination

AutonomousAttitudeTracking

AngularRateMeasurement

PartialImageDownload

StarCamera

X(X)

StarTracker

X X (X)(X)

AutonomousStarTracker

(X) (X) (X) X X (X) (X)

Key:X=Mandatory,(X)=Optional

TableRows:typeofstarsensors;tablecolumns:capability

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Annex D (informative)

Performance metrics applied to star

sensors

D.1 Overview

Thisannexdiscussestheperformancemetricsusedtoassesstheperformanceof

eachstarsensor capability. The definitions are derivedfromtheESA‐NCR‐502

(ESA Pointing Error Handbook) taking into account the specific case of star

trackers:

• themeasurementerrorsaresmall;

• theapproximationofsmallEulerangles

ispossible.

D.2 Application to Star Sensor measurements

D.2.1 Overview

Thisclauseapplies thestandarderrormetricdefinitionstothefollowingtypes

ofStarSensormeasurement:

• absoluteratemeasurements;

• inertiallyreferencedattitude,viaaquaternion;

• singlestarpositionmeasurement.

Thedistinctionbetweenquaternionandstarpositionmeasurementsismade.

D.2.2 Attitude quaternion measurements

The performance metrics AMEq, MMEq, RMEqandMDEq essentiallycapture

the various frequency ranges of measurement

 error sources that contribute to

theperformance.Thesearesummarizedin

TableD‐1.

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TableD‐1:Measurementerrormetrics

Metric LowerTimePeriodof

ContributionVariation

UpperTimePeriodof

ContributionVariation

AMEq

0

∞

MMEq(tMMEq)

τMMEq

∞

MDEq(tMDEqtOBS,MDEq)

τMDEq τOBS,MDEq

RMEq(tRMEq)

0

τRMEq

Typically,theseperformancemetrics,withappropriatetimeperioddefinitions,

can be used to constrain the following commonly referenced types of

measurementerror:

• Totalmeasurementerror–AME

q.

• Biaserrors‐MME

q.

• Longtermerrorsanddrifts‐MDE

q(withappropriatetimedefinitions).

• Shorttermerrors‐MDE

q(withappropriatetimedefinitions).

• Noiseerrors,orNoiseEquivalentAngle‐RME

q.

Each of the metrics can be used to constrain rotational or directional errors as

definedinclause

B.5.14.

D.2.3 Star position measurements

The performance metrics AMEs, MMEs, RMEs and MDEs essentially capture

the frequency ranges of measurement

 error sources that contribute to the

performance.Thesearesummarizedin

TableD‐2.

TableD‐2:StarPositionmeasurementerrormetrics

Metric LowerTimePeriodof

ContributionVariation

UpperTimePeriodof

ContributionVariation

AMES

0

∞

MMES(tMMEs)

τMMEs

∞

MDES(tMDEs,tOBS,MDEs)

τMDEs τOBS,MDEs

RMES(tRMEs)

0

τRMEs

Typically, these metrics, with appropriate time period definitions,can be used

toconstrainthefollowingcommonlyreferencedtypesofmeasurementerror:

• Totalmeasurementerror–AMEs.

• Biaserrors‐MMEs.

• Longtermanddrifterrors‐MDEs(withappropriatetimedefinitions).

• Shorttermerrors‐MDEs(withappropriatetime

definitions).

• Noiseerrors,orNoiseEquivalentAngle‐RMEs.

Each of the metrics can be used to constrain rotational or directional errors as

definedinclause

B.5.14.

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Annex E (informative)

Statistics

E.1 Confidence level

E.1.1 Overview

The performances have a statistical nature, because they vary with time and

fromonerealizationofasensor to another. Therefore, only an envelope of the

actualperformancescanbespecifiedandprovided.

This envelope is the combination of an upper limit and a performance

confidencelevel.

The performance confidence level

 indicates the proportion of the actual

performancesbelowtheupperlimit.

For example, the X absolute measurement error can be 10 arcsec with a

performance confidence level of Pc=95%. This means that the actual errors

fromonesampletoanotherarebelow10arcsecfor95%

ofthecases.

NOTE Performance confidence level is usually 99,7%

(corresponding to a 3 sigma values for Gaussian

distributions).

E.1.2 Accuracy on the confidence level

The verification of the specifications can only be done on a limited set of samples of the

whole statistical population:

• Onalimitedtimespan

• Onalimitednumberofsensors

The larger the set of samples, the better the knowledge on the performance

confidencelevel(P

c).

This implies that the actual confidence level is not perfectly known, but is

estimated with a certain accuracy ΔP, also called accuracy on the confidence

level.

Thisqualitativenotioncanbemathematicallyexpressedbyusing:

• The performance confidence level (P

c): it applies to the performances

quotedbymanufacturersandspecifiedbycustomers(usuallyas3sigma

values).

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• And the estimation confidence level. It applies to the estimation of the

performance confidence level (defined above). It represents the

confidencethatthesampleisrepresentativeoftheoverallensemble.

If not specified, confidence level means performance confidence level, and is

denotedP

cinthisdocument.

Theconfidenceestimationaccuracy(

)beingfixed,theminimumnumberof

samples(N)dependsontheestimationconfidencelevel.

• For an estimation confidence level 95%, then the minimum number of

samples is given by

)1(4

−

= . It means that if the number of

samplesislargerthanN,thentheactualconfidencelevelliesintherange

[]

PPPP

Δ− ; in95%ofthecases

• Foranestimationconfidence level 99,7%, then the minimum numberof

samplesisgivenby

)1(9

−

= . Itmeans ifthenumberofsamples

is larger than N, then the actual confidence level lies in the range

[]

PPPP

Δ− ;

in99,7%ofthecases

Furtherdetailscanbefoundinclause

B.2.

NOTE E.g. If the performance confidence level is 99,7%

andthe accuracy is ΔP = 0,1%, thenatleast 11964

samples are considered to actually demonstrate

that the actual performance confidence level is

between99,6%and99,8%(i.e.itisknownwithan

accuracyof

0,1%),withaconfidenceof95%.

E.1.3 Mathematical derivation

Nsamplesofarandomvariablex  froma probabilitydistributionfunction p(x)

are considered. Denote the actual performance confidence level of interest

P , with true value

x . Then thenumber of samples

N  within the set N

lying below

x  is sampled from a binomial distribution with mean and

variancegivenby:

NPNMean

=)( 

NPPNVar

CCC

)1()(

−

= 

Theestimate

oftheperformanceconfidencelevelat

x isgivenasfollows:



Therefore the mean and variance of the estimate

 of the performance

confidencelevelisgivenby:

PPMean =)

( (i.e.themeanvalueoftheestimateistheactualvalue)

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PVar

)1(

)

(

−



Now,let

PΔ

betheestimationconfidenceaccuracy,suchthattheactualvalue

c of the performance confidence level lies in the range

[

]

PPPP

Δ+Δ−

;

,

withagivenestimationconfidencelevel.

Thevariationsof

aresupposedtofollowaGaussiandistribution.Withthis

assumption,iftheestimationconfidencelevelissetto95%,(whichcorresponds

()

PVar

2± ), then the minimum number of samples in the set N to be

calculatedis:

)1(4

−



For a 99,7% estimation confidence level on

, the formula becomes

)1(9

−

, because 99,7% corresponds to a 3 sigma value for a

Gaussiandistribution.

Moregenerally,

)1(

PPn

CCC

−

foranC‐sigma estimation confidence level

of a Gaussian distribution.

NOTE For example, if the performance confidence level

on the error is 99,7% and the accuracy is

ΔP=0,1%, then at least 11964 samples are the

minimum number of samples used to actually

demonstrate that the actual confidence level is

between99,6%and

99,8%(i.e.itisknownwithan

accuracy of 0,1%), with an estimation confidence

levelof95%.

E.1.4 Minimum number of runs with no failure

ThepreviousclausefocusesontheminimumnumberNofsimulationstorunto

demonstratetheperformanceswithinagivenperformanceconfidenceleveland

agivenaccuracyontheestimationconfidencelevel.

Anotherapproach, more efficient from the implementation point ofview,isto

considerthenumberN

ofsimulationstorunifnofailureoccurstodemonstrate

the same performances. In this context, a failure is a simulation in which the

performanceleveltobedemonstratedisexceeded.

ThisnumberofsimulationsN

isusuallymuchsmallerthanN,whichmakesthe

approachmoreappropriate.

NOTE E.g.iftherequirementisspecifiedat99,73%,then

the number of samplesto estimate this

performance confidence level with a 95%

estimation confidence of the real value being

within ±0,1% of the

estimate is N=11964.

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However assuming no failures are seen, only

=1108 runs are required to prove that the

probability of failure is <0,27% to a 95%

estimationconfidencelevel.

Suppose that we have a specification P(x<x

max)>Pmin, with some statistical

interpretation. This means that for the system to meet its specification, in any

given trial the probability of having x<x

max is at least Pmin. Equivalently, the

probabilityofhavingafailure(x>x

max)inanygiventrialislessthanPf

max

=1‐Pmin.

Given a Monte Carlo campaign with N

 runs, ofwhich nf of these are failures.

Assuming that the real underlying probability of failure (not known to the

experimenter) is P

f, then the probability of observing nf failures in N

 trials is

givenbythebinomialformula:

(

)

()

(

)

nNn

NPnP

-1

!-!

′



The relation

()

(

)

()

ABP

BAP || =  with the normalization condition

()

∫

= 1| dABAP

yields:

()()

(

)

(

)

1-1

!-!

∫

′

dPPPPP

nPnNn

fff



Ifthereisnoa‐prioriinformationabouttheprobabilityoffailure,thenthemost

conservationapproach is toassumethatthe probability offailureis uniformly

distributedbetween0and1:

(

)



Thisyields

()()

(

)

1-1

!-!

∫

′

dPPP

nPnNn

fff



Then,foragivennumberofobserved failuresn

ftheprobabilitydistributionof

fisfoundtobe:

()

(

)

()

∫

-1

dPPP

NnPP

′



IftheprobabilityoffailureislessthansomevalueP

max

,thespecificationismet.

(This is equivalent to a minimum probability of not failing the specification.)

Givenn

ffailuresin

′

trials,the confidenceofthe specificationactuallybeing

met(i.e.ofP

freallybeinglessthanPf

max

)is:

()

(

)

()

1-,1,

-1

≤

max

∫

max

′

fffinc

nNnP

dPPP

PPprobC



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whereβincistheincompletebetafunctiongivenby:

()

∫

−

dttt

bax

inc



Thisfunctionisavailableinusualengineeringtools.

Usingthis formula,itis possibleto work outthe minimum numberof runs in

ordertomeetthespecificationswithagivenprobabilitytoagivenperformance

confidencelevel.

TableE‐1givesnumericalapplicationsforvariouscases.

TableE‐1:Minimumnumberofsimulationstoverifya

performanceatperformanceconfidencelevelP

Ctoan

estimationconfidencelevelof95%

MinimumnumberofrunsfornumberofobservedfailuresN

fail

Performance

confidencelevelP

C

fail=0 Nfail=1 Nfail=2 Nfail=3

68% 7 12 17 21

95% 58 92 123 152

99,73% 1108 1755 2329 2869

NOTE:‘failure’inthiscontextmeansviolationofthespecifiedbound,x>xmax.



There is no equivalenceto the estimation confidence accuracy

PΔ introduced

inclauseB.1.3.Itmeansthattheestimationconfidencelevelisatleastthelevel

specified(e.g.95%inthetableabove).

E.2 Statistical interpretation of measurement error

metrics

EachofthemetricsdefinedinclauseB.5istypicallyspecifiedandusedwithan

associatedconfidencelevel.

Anyperformancemetricsdependsonseveralvariables:

• thetimet;

• therealizationofthesensor(involvingthemanufacturingprocess);

• the observation conditions in which the performances are obtained

(e.g.angular rate applied on the sensors, orientation with

 respect to the

celestialvault).

Asitisnotpossibletobuildarepresentativesamplesetofsensors,thenotionof

statistical

 ensemble is used. A statistical ensemble of sensors is defined as a

collectionofsensorsrepresentative ofthemanufacturing process, in whichnot

allsensorsarenecessarilybuilt.

Because a metrics depends on several variables, there are several ways to

interpretaspecificationanditsconfidencelevel:

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• Temporalinterpretation

⎯ The worst case combination of sensors and observations is

considered.

⎯ The worst‐case sensor/observation combination is defined as the

worst‐case sensor observing the worst‐case direction in the

celestial vault under the worst‐case observation conditions. The

worst‐casedirectionistheoneleadingtotheworstperformanceof

the sensor. It is related to the worst distribution of stars

 over the

starsensorfieldofview,takingintoaccountembeddedalgorithms

andcatalogues.

⎯ Theperformancesareestablishedwithrespecttotime.

⎯ The specification metric is ‘less than S for n% of the time for a

worst‐case sensor/observation from a

 statistical ensemble of

sensors/observations’.

• Ensembleinterpretation

⎯ Astatisticalcollectionofsensorsisarbitrarilychosen.

⎯ Agivensetofobservationsisarbitrarilychosen.

⎯ The time is set to the worst case time, i.e. when the performances

obtainedforagivensensorandobservationareworst.

⎯ Thespecificationmetricforthistype ofvariabilityis‘lessthanthe

level S in confidence level n% of a

 statistical ensemble of

sensors/observationsfortheworst‐casetime’.

• Mixedinterpretation

⎯ The mixed interpretation combines the ensemble and temporal

variationtocapturetheerrorvariabilitybothovertimeandacross

theensemble.

⎯ Thespecificationmetricforthistypeofvariabilityis‘forarandom

sensor/observation from the statistical

 ensemble,and at a random

time,themetricislessthanSwithaprobabilityofn%’.

For a generic measurement error source with an amplitude and a time

variation, the ensemble interpretation gives the distribution of the error

amplitude over the statistical

 ensemble of sensors/observations, while the

temporal interpretation covers the error variation over time for the worst‐case

amplitude.

For the AME, RME and MDE metrics defined in clause

B.5, the statistical

interpretation can in principle be ensemble, temporal or mixed. However, the

nature of the MME metric means that only an ensemble interpretation is

appropriate. Specific identification of the interpretations to be used in this

specificationisgivenin

AnnexD.



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Annex F (informative)

Transformations between coordinate

frames

Transformationsbetweenanytwoco‐ordinateframes,AandBcanbedescribed

by the transformation matrix

−

 which transforms the components of a

vectorfrom‘B’frameto‘A’frame:

rTr

−



where

r arethecomponentsofthevector r inthe‘A’frame,and

r arethe

componentsofthesamevector

r inthe‘B’frame.

The discrepancybetween both frames ‘A’ and ‘B’ is defined by 3 Euler angles

around 3 distinct axes. In this Standard, the rotations are always small,

thereforetheorder of the rotations is notimportantandthese rotations canbe

takentoberotationsaroundthe

X‐,Y‐andZ‐axesofeitherframe.

Thetransformationissimply:

⎥

⎦

⎤

⎢

⎣

⎡

φΔ−θΔ

φΔψΔ−

θΔ−ψΔ

≈

−



where

Δ ,

 and

 are the 3 small rotations respectively around X, Y

andZaxestransformingthe‘B’frameintothe‘A’frame.

Thediscrepancybetweenbothframes‘A’and‘B’is:

⎥

⎦

⎤

⎢

⎣

⎡



Thediscrepancyisafunctionofthetime.

NOTE Theperformancesofstarsensorsaremeasuredby

applying the metrics defined in

Annex D to this

vector

.

For star sensors, this vector typically represents the angular errors between a

measuredquantityanditsactualvalue.

NOTE E.g. With ‘A’ frame being the actual star sensor

frame and ‘B’ frame being the measured star

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sensorframe,then

 represents themeasurement

errorsofthestarsensor(see

FigureF‐1).

rotation

Original Frame

After 1

rotation

After 2

rotation

Final Frame

FigureF‐1:Anglerotationsequence

In this case the 3‐axisEulerrotation parameterization corresponds to rotations

aroundtheB‐frameaxes.

The separation of two frames A and B, defined in the ESA Pointing Error

Handbookandwrittenas

(

)

Tsep

−

isdefinedas:

()

⎥

⎦

⎤

⎢

⎣

⎡

−

Tsep



Thisfunctionrepresentsthediscrepancybetweenthetwoframesandisusedto

measurethestarsensorperformances.

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Annex G (informative)

Contributing Error Sources

G.1 Overview

ThisannexlinkstheerrorcontributorstothedefinitionsderivedfromtheESA‐

NCR‐502 (ESA Pointing Error Handbook). The traditional contributors and

performances are compared with generalized error with respect to the

correspondingcorrelationtime

τgivenforeachcontributor.

TableG‐1:Contributingerrorsources

Errorcontributors Comments

Bias

‐ on‐groundcalibrationresidual

‐ launch‐inducedmisalignment(vibrations,

depressurization,gravity…)

BRFvsMRFmisalignmentduetoafter‐launch

ageing

MME(

τ=infinite)



MME(

τ=lifetime)

Thermoelasticerror

BRFvsMRFstabilitydueto:

‐ stabilizedopticalheadtemperature

‐ gradientcausedbyconductiveand

radiativeeffects

MDE(

τ=oncethethermalscenarioisknown.)

τ =correlationlength

τ obs=observationlength

FOVspatialerrors

‐ PointSpreadFunctionvariabilityacross

theFOV

‐ residualofcalibrationoffocallength

(includingitstemperaturesensibility)andoptical

distortions(includingchromatism)

‐ residualofaberrationoflightincase

whereitiscorrectedatquaternionlevelandnotat

starlevel

‐ CCDCTEeffect(includingits

degradationsdueto

radiations)

‐ catalogueerror(includingstarproper

motionandparallax)

Theamplitudeoftheseerrorsareindependentofthe

rate.

The

τisassessedbythesupplierintheangular

domain.

Thereisaneedtogetthefiguresforseveral

τ values.Theuseofautocorrelationfunctionof

spatialerrorisrecommended.

MDE(

τtobedescribed)

Canbeconvertedbytheuserintimedomain

dependingonthespecificapplication.usingangular

rate

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Pixelspatialerrors

‐ detectornonuniformity(FPN,DSNU

(DS(T),radiation,integrationtime…),

PRNU(straylight,starsignalphotonicnoise)…)

‐ centroiding(ratedependent)

The

τisassessedbythesupplierintheangular

domain.

Canbeconvertedbytheuserintimedomain

dependingonthespecificapplicationusingangular

rate.

MDE(

τlinkedtopixelFOV)

Temporalnoise

‐ starsignalshotnoisedependingonstar

signal(StarMagnitude,exposuretime,optical

contamination,transmissionloss,defocus,rate…)

‐ backgroundsignalshotnoise(straylight

level,detectortemperature…)

‐ read‐outnoise

‐ quantificationnoise

‐ datationnoise

RME(

τ=0orlessthanthesampletime)

Aberrationoflightorresidualofaberrationof

lightcorrectionifcorrectedatstarlevel

MDE(

τ=TBDbyuser)

residualofaberrationoflightcorrectionifcorrected

atstarlevel

Asthiserrorisverydeterministic,itispossibleto

correctitinsidethestartracker‐supposingthatthe

velocityinformationisgiventothestartracker.A

fewcasesarequoted:

1) acorrectionis

performedforeverystar

direction,

2) auniquecorrectionisperformedgloballyfor

auniquedirection(example:lineofsight,or

barycentreofthemeasuredstars)andappliedonthe

quaternionoroneachstarmeasurement,

3) acorrectionisperformedonlyfortheEarth/

Sunvelocity,

4) nocorrection

isperformed.

Dependingonthecorrection,theerrorresidualis:

‐ aFOVspatialerrorifthecorrectionis

performedglobally(case2)

‐ anorbitalerrorinthecase3(dependingalso

ontheattitudeofthespacecraft)

‐ alongtermerror(oneyear)+orbitalerror

forthecase

4.

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Annex H (informative)

Example of data sheet

H.1 Introduction

The data sheet in Figure H‐1 shows an example of data sheetfor autonomous

startracker.

Thefieldsthatcanbefilledinareidentifiedinanitalicfont.

Theexamplevalues filledin arejustforformattingpurposesanddonotrelate

toanexistingstarsensor.

H.2 Rules applied

ThefollowingruleshavebeenappliedtoprovidethedatasheetinFigureH‐1:

• useofthecontentoftheexampledatasheetproposedinthe“StarSensor

Terminology and Performance Specification Standard”, issue 1 and

additionofsomekeyitems.(firstversionofthepresentdocumentissued

byESAstudies);

• the data sheet has been limited to one page

 of format A4 but is not

mandatory.

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Detailed Data Sheet

Companies Logo

Name: Name as supplied by manufacturer

Type:

Autonomous Star Tracker

photograph of the unit

Configuration:

Single Box, Single Head

Specification: Detector:

STAR1000 APS

FOV:

20 x 30 deg rectangular

Interface:

MIL-1553

Power:

<10,0 W

Voltage

Output Frequency:

10 Hz

operational

-10/ +20

°C

Mass (Baffle):

1,0 (1,0) kg

Dimension:

120 120 225

(including baffle) width depth height

EOL performances mission dependent

BOL performance

quoted @ worst case temperature / celestial bodies aspect angles

quoted for normal to LOS and for along LOS

Aspect Angles:

Sun Exclusion Angle:

25 deg (@ 1AU)

Earth Exclusion Angle:

20 deg (illuminated Earth limb at 70 km)

Moon Exclusion Angle:

n/a deg (full Moon accepted in the FOV)

(normal to / along LOS) statistical distribution

Bias

5,0 / 15 arcsec max

TBD

thermal stability

0,1 arcsec/K

uniform

spatial error (Pixel)

2,0 / 4,0

arcsec p/p arcsec TBD

spatial error (FOV)

1,0 / 3,0 arcsec p/p

TBD

Attitude tracking can be performed under the following conditions:

statistical distribution

Rate

0,0 1,0 5,0

deg/sec

Acceleration

0,0 0,1 1,0

deg/sec

temporal noise

10 / 70 14 / 98 25 / 175

arcsec 3

gaussian

Prob. of correct

≥

0,9999

≥

0,9990

≥

0,9500

P (10s, random directions)

Prob. of false attitude

≤

0,0010

≤

0,0030

≤

0,0100

P (10s, random directions)

Maintenance level of 999s/1000s 990s/1000s 999s/1000s

# of false stars

0 10 50

P (10s, random directions)

≥

0,9999

≥

0,9950

≥

0,7000

Prob. of correct attitude determination

P (10s, random directions)

≤

0,0010

≤

0,0050

≤

0,0100

Prob. of false attitude determination

Track

Maintenance

/Period

999s/1000s 990s/1000s 999s/1000s Maintenance level of tracking

ecial Features:

• Tolerance u

to 1000 SET/cm

/s at initial ac

uisition

• Tolerance up to 4000 SET/cm

/s in tracking

• Life time 15 years geo-synchronous orbit (FITS number)

• High-rel EEE-parts, ITAR free,

≥

100krad

• Attitude quaternion

• Angular rate and Acceleration limit for tracking : 1 °/s and 3° / s

• Time stamp

• Health status and star

osition & ma

nitude

• Image download / Code&Data down- & up- load

FigureH‐1:Exampleofdetaileddatasheet

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Bibliography

ECSS‐S‐ST‐00 ECSSsystem–Description,implementationand

generalrequirements

ESA-NCR-502

ESAPointingErrorHandbook

